Nothing here makes sense

He doesn't immediately berate him for his failure to purge the planet, he takes a moment to muse on how much Goku has grown and how he looks just like their father Bardock. Keep in mind, Bardock wasn't even a spark in Toriyama's mind, so for him to have Raditz say this when he's supposed to be a throwaway villain shows an unusual level of fondness for family, especially considering what we later learn about Saiyans

Raditz, upon realizing Goku doesn't remember him, isn't just annoyed that Goku forgot his mission, but seems distraught that his little brother doesn't remember him.

It's a small detail, but again, cements that Raditz shows an unusual level of attachment to family bonds, especially for a Sayain.

And then (and this is something important to keep in mind) he declares that he will find a way to recover his little brother's memories because Goku is NEEDED.

His priority isn't to get the planet purged or punish Goku for failing, it's to regain the only biological bond he has left, however little of it there may be.

I find it interesting Toriyama wrote this piece of dialogue. It just seems odd he would write such layered dialogue to characterize a villain he always intended to kill off ASAP. I guess it was to play into the whole "subverting the brother trope" but still, it doesn't make the characterization any less interesting

When Raditz tells Goku how their planet was destroyed and how everyone died, he AGAIN emphasizes that this means their parents died too.

Again, Raditz really seems to put value on his family. Note he says PARENTS not just father. Even Vegeta, for as long as we've gotten to know him, never talks about his father King Vegeta that much (if at all? He's thought about him, but not really talked about him) I find Goku's reaction interesting too, as if deep inside, despite not remembering Bardock and Gine, his heart still feels the pain of losing them. (Could it be possible Raditz noticed his reaction and took that as a sign that Goku felt the bond too, hence his following actions?)

Like @masakoxtra said, Bardock's line seems to be unusually empathetic for Saiyans. (He talks about it at 3:30)

Even Raditz, despite living his whole life under Frieza's boot and submitting to the bullying and callousness of Vegeta and Nappa and hardening his heart because of it, hasn't completely lost his sense of empathy, it's part of his nature albeit incredibly suppressed.

Raditz then has an unusually distressed response when he realizes Goku doesn't have a tail.

He doesn't mock him for losing it or immediately gets disgusted by his weakness, he is outright horrified and then gets mad at Goku for letting others just remove his tail (From Raditz's perspective, It would be like if Goku just let his arm get cut off to fit in with a race of one-armed aliens).

For Raditz, he views it as a form of betrayal, not just of his race, but the idea that his own brother would rather pass as a lowly earthling than embrace his own heritage (family being something Raditz clearly values) really gets to Raditz on an emotional level.

Now that I think about it, Raditz kinda goes through 4 out of 5 of the stages of grief for the brief time he's alive.

His first reaction is denial that Goku had forgot him and accepted life on earth, then anger that he would rather live as an earthling than be with his Saiyan kin, and then he starts the bargaining phase, trying to entice Goku with the idea of fighting saying that he's a Saiyan and it's in his blood.

When that bargaining doesn't work, he resorts to a different form of bargaining.

Blackmail.

Raditz steals Gohan trying to force Goku to join him. He tells Goku to kill 100 humans by tomorrow as proof of his submission, but pay attention to the wording:

Raditz says "when you decide to join us, and you WILL decide to" that's how much confidence he has in Goku's devotion towards his son EVEN THOUGH Goku's a Saiyan.

Raditz doesn't have a shadow of a doubt that Goku will do everything in his power to protect his son, even if he is a weak crybaby. Saiyans don't typically care much for their kin as shown in several flashback material later on (in fact it's later explained that they'll completely disown and abandon babies that are too weak to be considered useful. They have a very Spartan-esque society).

But Raditz knows he can use Gohan as leverage because Raditz actually understands emotional connections between family members, something he would've likely valued all the more being considered weak himself.

A lot of times people are able to use emotional manipulation because they either understand or were a victim of similar manipulation.

He then warns Goku that he might as well comply because everyone is going to die anyway, the earth being scheduled for purging. He hammers home the point that Goku's defiance is pointless and he really doesn't have a choice anyway so he may as well submit.

But what Raditz is doing here is almost an act of compassion (for a Saiyan). The way he sees it, Goku will die if he doesn't comply, so joining them is the only way he'll be able to survive. If he didn't care about Goku's life, why warn him? Why give him a chance to prove himself?

In fact, why would Raditz need Goku to prove himself when he was willing to take him without that before?

This is just an idea, but could it be...because of his scouter?

Remember, his scouter was open the entire time so Vegeta and Nappa are listening in. If Vegeta was listening it, after hearing about Goku's weak power level and his defiance and kind-nature, Raditz probably knew Vegeta might just dispose of Goku when they returned, considering him a disgrace to the Saiyan race. So Raditz has to have Goku prove himself by killing a bunch of humans to show Vegeta he's worth keeping alive.

It's horrific in Goku's eyes, but to Raditz, the lives of a few humans is inconsequential compared to his brother. This again is why Raditz says Goku has no choice, Vegeta won't give them a choice.

This also might be desperation on Raditz's part. If we are to consider the opening of Dragonball Z: Kakarot canon, Vegeta and Nappa mock him, Nappa going as far as to declare it's why he's called "Raditz the Runt", apparently a knickname he's saddled with in the Frieza force.

Raditz, instead of responding angrily or protesting as most Saiyans would, bows his head and pathetically apologizes, promising things will be different next time, showing that not only is this bullying common, but Raditz has just accepted it at this point. The way Raditz treats Goku when meeting him may stem from this treatment, he's trying to sway his brother the only way he's seen, through brute force and intimidation.

But Vegeta, getting sick of Raditz's weakness, promises to kill him if he screws up again, and if Vegeta promises death, you know it's coming. Raditz, panicking as his self-preservation instincts kick in, mentions his brother, saying he can help make things easier, but really Raditz just doesn't want Vegeta to kill him. Even then, Vegeta scoffs "The fact that he's YOUR brother doesn't exactly fill me with confidence" It's possible that Raditz did actually forget his brother and it was only in his panic, scrambling mentally for any way to save his life, that in that moment of desperation he at last remembered Kakarot.

Again, if we are to consider this conversation canon, Raditz needs Goku to survive to better the odds of his own survival, it's only after he's in a pod heading to Earth that he has time to think about Kakarot and wonder why he hasn't tried contacting them after so long.

But back to the OG manga, After Raditz gives Goku his ultimatum, he says this:

Again, he could've stopped at "I hope you don't disappoint me" but to follow it up by emphasizing it's for both his and Gohan's sake is noteworthy.

And even though Raditz clearly doesn't have much of a connection with Gohan as he does with Goku, I find this bit particularly interesting:

He barks at Gohan to stop crying and states that he possesses the proud blood of Saiyans. Yes, he is annoyed by Gohan's crying, but he also feels that he's better than that since he is still a Saiyan and wants him to be strong.

I like to imagine that Raditz is repeating something Bardock told him when he cried as a child, it feels like a very Bardock thing to say.

I particularly like the english dub of this scene, Justin Cook gives such an interesting and tender delivery of the line.

Also I really like how Raditz pauses to look at Gohan before walking away in the anime, I like to interpret it as Raditz seeing a bit of himself as a child in Gohan, but quickly burying those feelings.

There's a little fancomic I found that really drives that idea home.

When Goku and Piccolo show up, before they even fight, Raditz says this:

Again, why warn Goku? This feels more like Raditz is still in the bargaining mindset, he's trying to get his brother to give up and now must resort to brutally beating him to get his point through.

And then followed by this.

Remember, his scouter is open, so it's entirely possible he's acting ruthless and declaring they'll die so he won't look soft to Vegeta. I mean, he'd kill Piccolo without a thought, yeah, but Goku...? It may still be a bluff.

Plus, if he was serious about killing them, why stand around and let them plot instead of finishing them off?

The tail scene is where we see Raditz's cowardly nature on full display. But I think this moment really enhances his character because most Saiyans probably wouldn't beg for their lives, at least not to the degree Raditz is doing, they're too proud a race.

Raditz starts rambling about how he'd never actually kill his brother and his death threats were just bluffs.

Yes, we know it's a ploy to get free, but could there be an iota of truth in there? The fact he could've cut off his tail but was waiting for Piccolo to fire off his second Makenkosopo shows that Raditz is a quick thinker and very calculating.

Plus he probably didn't want to have to lose his tail unless he absolutely HAD to.

Goku was NOT stupid for letting go.

After Goku releases him, Raditz mocks his softness stating that he, a Saiyan-warrior wouldn't hesitate to kill their own brother, only to confusedly ask if Goku wants "a demonstration".

Like, if he wouldn't hesitate to kill his brother, why is he hesitating to kill his brother?

He's not killing him, he's torturing him, he could easily end it.

Remember that Double Sunday he shot off with ease earlier?

And of course after Raditz and Goku get turned into donuts, Raditz says one of the saddest lines in retrospect:

Like, he is relying on Vegeta and Nappa to save him, believing that they'll value him as a Saiyan and bring him back because HE HAS NO ONE ELSE TO RELY ON.

Right before he dies, he's in a sort of stage 4 depression where he can't believe this is how his life is going to end, dying alone and disgraced on some backwater planet at the hands of his own brother, their family line coming to a miserable end. He's never allowed to come to stage 5: acceptance (which is often where the change in a person's perspective/character tends to happen) because he dies and is forgotten.

Another thing that makes me sad Raditz didn't survive is cuz he's the perfect medium between Goku and Vegeta.

Goku rejects his saiyan heritage while Vegeta clings to it, but Raditz feels like he could easily straddle both worlds. He'd cling to his saiyan heritage out of love and respect for his parents (He'd still call Goku Kakarot, not because "it's a Saiyan name" like Vegeta, but because it's the name Bardock and Gine gave, his reason a much more personal one).

However, Raditz would have plenty of things NOT to like about Saiyan society, especially with how he and his father were treated as low-class warriors.

Being on earth, surrounded by kind people who don't belittle him and show basic kindness and respect would quickly endear Raditz to earth (remember, Bardock's kin are unusually empathetic for Saiyans).

Plus, being around Goku, who'd no doubt encourage and be proud of Raditz whilst training, would do a lot to boost Raditz's confidence (Goku looking like Bardock a way to ease his yearning to prove himself to his father) and further make him enjoy earth.

I like to imagine that, while Goku always wears a training gi from earth and Vegeta always wears some semblance of saiyan armor, Raditz would probably have a saiyan breastplate resembling Bardock's (as a kind of tribute to his dad) and go with loose pants like Goku which is good for training, visually symbolizing his willingness to find the balance between two worlds.

If Raditz had survived in the canon, this could've played even further into Vegeta's sense of isolation post-Cell arc. During his whole Majin Vegeta vs Goku fight speech, he could've said something like "And imagine the frustration I felt, when the only other pure-blood of my race left, your brother, that low-level trash who'd trembled for years under my elite warrior might, not only obtained the power of a super saiyan, but deemed me, ME the prince of all Saiyans UNWORTHY of his time! UNWORTHY FOR HIM TO FIGHT!"

Oh, and...