Wrote up an explanation on why I'd been ruminating on Yukio having AVPD a while back and realized I never posted it (or the OCPD one I wrote up some time after), so here we are!
For starters, his self-isolating behaviour and preoccupation with feelings of inferiority were one of the things that kinda had me looking at him through an avoidant lens. According to the DSM, patients must have "a persistent pattern of avoiding social contact, feeling inadequate, and being hypersensitive to criticism and rejection".
Reviewing the criteria (of which a mininum of 4 points is needed), I think Yukio's symptoms would revolve around the following 5:
Unwillingness to get involved with people unless they are sure of being liked
Reserved in close relationships because they fear ridicule or humiliation
Preoccupation with being criticized or rejected in social situations,
Inhibition in new social situations because they feel inadequate,
Self-assessment as socially incompetent, unappealing, or inferior to others,
Yukio does a lot to avoid being seen as weak, and doesn't have many - if any - close relationships. Rin is probably the closest to a peson he's ever been and even then, there is so much that he has to hide that their level of closeness with each other is very uneven. Rin defintiely thinks they're closer than they are.
Self-isolation is also a common tactic, especially when under stress. This tactic can backfire though because it sort of proves their feelings, especially if other people don't notice or question the isolation. However, like with Yukio, pointing out the isolation can backfire because as much as they may want to be seen, being seen means that people know they're worthless and that's not good either.
People with AVPD often feel they cannot rely on other people, or that their problems - no matter how painful they may be personally - are not enough to consider reaching out to someone over. Not to mention reaching out for help would only reinforce the preexisting feelings of worthlessness. For that, they will often avoid treatement, and, like with many PDs, are typically diagnosed when seeking help with other issues (like depression). Yukio's reluctance to seek help - even when encouraged by Renzou or others - is a good proponent of this. People can't know that he's struggling because if they do, they'll know he's weak and that is a sentence worse than death.
They may long for social connections, but struggle with being able to offer themselves up to other people. While some may choose to deal with this struggle by avoiding contact with people altogether, others, like Yukio, may present a competent and aloof persona. They don't need help, they are the helper.
This is kind of what I mean by non-stereotypical/atypical presentations. I think when you first consider AVPD, you think of someone who is overly social anxious and extremely timid, in the same way that people may think someone with ASPD must secretly be a serial killer or whatever. But people are complex and no presentation is ever gonna be the same (hence the proposed used of the AMPD/the ICD-11 doing away with labels altogether).
For example, with Leo (PJO character I also headcanon to have AVPD), he purposely presents himself in a way that causes rejection because if it's going to happen anyway, he might as well be in control of it by being as annoying as possible. At least that way he's not rejected for who he really is.
With Yukio, he's not timid anymore. He's cool and collected. He projects maturity and strength and gets unfurled whenever someone points out or comes close to discovering that he's not. Internally, he is a mess, but externally he is calm and he is capable. He doesn't think of himself as being superior and I don't necessarily think he wants people to think of him like that. But I think he's easily envious of others, especially when it comes to areas he thinks himself low in, and so doubles down on avoidance by ensuring there's always enough space between him and other people. Someone pointing out that space just leads him to feeling worse and doubling down on the space even more.
I think the other end of it is that Yukio is likely more helpful, or willing to offer help, than is healthy for him. Part of it is to prove that he's not useless and part of it is to gain a sense of socialization without offering too much of himself. I think about his relationship with Shiemi. They both clearly think of themselves as friends, but, like with Rin, Shiemi probably thinks (or at least thought prior to the whole child of Satan reveal) that she's much closer to Yukio than they ever were. Yukio, however, is aware of this distance and probably more comfortable with it than he'd like to be. But there's a lot of reasons for him not to be open with her, so it's fine, probably, he's not gonna think about it too much. Still he's willing to help her study because it makes him feel useful and gives him a reason to hang out with her.
Honestly, Rin's quick rapport with her likely furthered his feelings of inadequacy, especially in comparison to Rin.
Oh, yeah - with Yukio, he's not unaware of the fact that he does excel in a lot of areas, but it's never enough. That's also a common theme I've seen being discussed by people with AVPD. It doesn't matter how much they do or how well they do, they will never be enough. Not necessarily in comparison to anyone else, but just in general. They could solve world hunger and achieve world peace and it still wouldn't be enough.
Yukio, I think, likely experiences that a lot. He knows he's smart, but it's not enough. He knows he's good at his job, but it's not enough. He knows he's strong, he's knows he's a good fighter, but it's not enough and it will never be enough to make him feel like he deserves a place in his own life, much less in the lives of others.