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Both in a Watsonian and Doylian sense, it's cultural.

He's from a Japanese series, and iirc saying I Love You to your wife and kids isn't really a thing in Japan. You might notice that Goku's not riddled with pride, but he doesn't say it either (the closest he gets in the manga is saying that he "loves that part" of Chichi when Vegeta mentions Saiyans like strong women). Both in-universe and beyond the fourth wall, culturally they're more about actions than words.

From a strictly in-universe standpoint, while there are plenty of cultures like that here on Earth, Vegeta is not from here. He's not human, and he was raised in a military -- verbal affection is never going to be instinctive for him. He expresses his affection by trying to keep his loved ones safe (keeping Kid!Trunks out of serious battle, silently putting his body between Future!Trunks or Bulma and anything he perceives as a threat are great examples of how he publicly expresses his love for them.) Many cultures treat nonverbal affection as normal, and there are even some cultures that express love through comfortable silence, and I think that sharing space without needing to fill it is something more in Vegeta's wheelhouse.

Just wanting to be physically near his family when he could be literally anywhere else is an expression of love -- which is why I'll die mad that they changed this scene in the anime.

Pride certainly has something to do with it at the beginning, but imo it's less I'm Too Good For This and more This Is Not How Things Are Done (although depending on what era of Vegeta we're talking about it's probably a mix of both). Vegeta spends the whole first series unlearning cultural propaganda that ultimately did nothing but hold him back, and by Super('s manga) he's cooing over his new baby in front of God and Everybody.

tl;dr: Vegeta tells his family he loves them all the time, and they do the same for him, they just communicate it in less direct but more substantial ways. Plus, like I said in another post, talk is cheap when you're raised by liars. "I love you" wouldn't mean anything to him unless it was earned first anyway, and by then it's just an 'I know' for everyone involved.

I guess Ginyu would be more annoyed that their dance moves are actually better than him and he can't accept that he got bested

Also since it's a Lion king AU,

That scene still gives me chills after 20 years

HOPE!!

They established that Saiyans get there growth spurt at age 15. Vegeta is 5 years older than Goku so he’d be 16-17 when he’s summoned in this scenario, meaning he already got his growth spurt before this hypothetical scenario of Shenron summoning him for Bulma.

Vegeta and Bulla

A Bulla redesign based on this tweet:

Oh yes. I think since Buu Vegeta’s been working at being a Dad by Earthling standards. I’ve mentioned before that he’s never going to be the Buddy Dad like Goku, but he knows how to play his son’s video games and the very first thing Bulma did with Bulla was go to hand her to Vegeta, which IMO tells me that he’s been much more open to hugs and holding and casual affection with Trunks since they broke through that barrier (and his response to that was to reach out with both hands and embrace her, before he was reminded of the situation).

Vegeta: I am not the dramatic one here. I have never been dramatic.

Server: Sir we don’t have Pepsi, would you like a Coke?