As someone who does know a good bit about skating lemme just add to this point by point.
First:
He was in the final. It's absolutely correct that that would basically make him 6th in the world even if the Grand Prix is technically not Worlds. He is good. This is the type of event most skaters, even on an international stage and in the top of the top, never reach. It's really impressive. I've talked at length about it here.
Something I didn't mention in the post was the GP qualifiers and how they're decided. I won't go into specifics but Japan always hosts a qualifier known as NHK Trophy and with hosting comes the privilege of getting a couple free spots (exact number depends on discipline) in the lineup to debut skaters in the GP or give a skater only assigned once otherwise a second assignment. The thing to note here is that Yuuri is not assigned to NHK. In figure skating language this means that Yuuri definitely earned both his assignments purely of his own merit, no NHK advantages needed.
Second:
He's Japan's Ace. Multiple things here:
1) Having a nickname that prevalent is an achievement of its own. There aren't many who have nicknames which leave smaller skating communities and him being Japan's Ace seems like a universal nickname. That's his title everywhere. I'm sitting here trying to think of all the skaters I know who have sticking titles like this and my list is currently quite short.
2) He's Japan's ace because he's the best skater in the country. This is confirmed by the fact that, well, we don't hear of any other Japanese skaters in the GP. They might've just been left out for time but still, if they were good enough, they would've been considered someone to watch out for and have been mentioned.
Another confirmation is that on the JSF website, Yuuri is the skater on both figure skating banners. If they had literally anyone else to put there, they would've* since this is basically the JSF admitting that the Japanese figure skating field is very shallow. Not even just men's, they could've put a woman, a pair, ice dancers, anyone, they could've put goddamn Minami but they didn't. They do not have anyone else this big. He is the top of Japanese figure skating. Oh, and he's also the only men's skater listen in top certified athletes on the JSF website.
*I imagine there's some fan service here, people wanna see Yuuri. The other banners are just the same pic which was probably for time but they took the time to draw two Yuuri's when they, as shown above, have two women's skaters with faces right there. Just draw any woman skating, slap the given hair onto them and bam, you've got another skater to put on the banner. Either the JSF are doing everything they can to market figure skating as a men's sport to attract male skaters or the women here are not even close to Yuuri's level (which sounds unlikely but anyway, Yuuri being on two banners was a deliberate decision).
Third:
His trademark elements. Let's just go in order:
The triple axel is, to put it midly, really fuckin hard. It's 3,5 rotations in a split second and the only triple jump categorised alongside the quadruple jumps because of its difficulty. Sure, it's everywhere in the top of men's figure skating now but this show took place 7 years ago, at the start of the quad revolution (this probably happened a little earlier in universe but still) and while 3A's were very much still commonplace in the top, they weren't seen how they are now.
SO, take this information and then add that it's his signature jump. He's known for his ability to perform it beautifully. Sure, the 4A is Ilia Malinin's jump just because he's the only one in who can really land it but Yuuri doesn't have that luxury, the 3A was first landed 1987. It's his signature for the reason the 2A is Kaori Sakamoto's. He's just that good at it.
Step sequences are in large part the mark of artistry in a program. There's a lot more to it, but if your step sequences are gorgeous, you've usually got the other stuff down as well but if you're so good at steps that you're known for it, you're a certified artist in skating. I wouldn't be shocked if Yuuri's the type to get level 4's on his step sequences (level 4 basically means it was very difficult and complex). Actually,
the element list from Yuuri's short program in Sochi shows that he got a level 3 on his step sequence (StSq3) and considering he barely landed his 3A and, ya know, wasn't doing so hot at that final in general, that's still pretty good. (Also, I will go insane if I hear someone say Yuuri did poorly at the Sochi GFP ONE MORE TIME, I SWEAR TO G-)
Spins are one of the harder trademarks to get, at least in my experience, since they're hard to judge as the average viewer. To the untrained eye, they all look really fast and incredible. Even people who I'd expect to be known for their spins, because they're consistently really good, aren't.
Then again, it can sometimes be a little random what people get as signatures since it's more what people latch onto, not necessarily that said skater is the best of the best at said element. But you need to be pretty good in either instance so yeah, his spins are excellent, trust that.
Fourth:
Viktor is impressed by his musicality. That's it. The best person in the sport, the most decorated figure skater ever, the 5 time World and GPF champion who outscored second place at the Sochi GPF by 34.30 points, basically admitted that Yuuri's musicality is beyond his. And Viktor doesn't suffer from an I suck mentality, he knows he's good but still thinks Yuuri's got the stuff to win. He knows good when he sees it.
Fifth:
Yuuri is the most unreliable narrator ever and should not write an autobiography. Every time he puts himself down, there's canon proof to diprove his point. Failed miserably at Sochi? Read this meta and the first point of this reblog. Dime a dozen? See JSF website, he's the only male skater in the top and the one on the banners. Give me any selfdegrading statement of his and I bet I can find canon proof of it being blatantly untrue.
Like OP said, Viktor also calls him out on this and Yuuri admits that he's insecure and thus unreliable in his view of himself. It's just straight up canon. No pussling it together required.
Sixth:
He knows. Yuuri might be insecure but even he can see that he's a very good skater, even if it's often interrupted by his anxiety. He wouldn't have made it as far as he did if he had no confidence at all. Friends, family, and coach can only do so much to push you to keep going and I love the Katsukis but I don't see them being the ones to keep pushing.
He might be known as "The Skater With the Biggest Glass Heart" (I can't remember the actual wording, feel free to correct me) but he sure ain't weak. He's canonically stubborn as hell and doesn't stop for even his anxiety. Get lost, brain chemicals, Yuuri's gotta go slay.
So uh, TL;DR: Yuuri is a very good skater.
