As I read it, it's not that she's incompetent or stupid, or that they're punishing her, or that if she just works hard enough they'll reward her. That's the whole misunderstanding on her part. It's that having watched her for years, she doesn't have what they're looking for, what they think she needs. Sometimes she'll be really helpful and successful and nobody can take that away from her, but she can't handle situations the way she needs to be able to handle them. So her never-say-die attitude, her willingness to try to learn from her mistakes, while good qualities, aren't going to give her what she needs to have.
I'm not dismissive of the sexism idea. I can see why it's obvious to think about. For me, I don't find it sexist. There are women/girls who don't have Steph's issues, there are men/boys also unsuited to doing this. Personally I *don't* accept the Year One treatment for Babs because I think it does try to make her into a feisty girl who's just trying to win validation from Batman and personal expression and that's totally not what Babs was and I think it's demeaning. But even in that story Babs was suited to doing this.
Like I said, I'm not dismissing people feeling like this is connected to her being a girl. I'm not going to come out and say there can't be any sexism at work here. But for my reading of this character it seems like they've pretty consistently showed me the problems that she has that keep her from reaching the next level, and so everyone else comes across as speaking the truth to her rather than just arbitrarily raising the bar so that she can't reach it. For me--and I know for others it reads differently so they can't enjoy it--but for me this isn't demeaning to Stephanie as a character, it doesn't make her a loser. It just puts her in a hard situation where eventually she has to deal with this being the truth. There really is nothing she can do to please them because it's not about pleasing them. It'd be interesting to see this played out in a boy or a girl--I think she's a girl so they could have a romance with Tim. I think Jason Todd was kind of a prototype for the possibility of this kind of story, though his problems were different from Steph's.
Anyway, that's just how the conflict always plays out for me, that Steph's unique in that she's somebody who wants to do this but isn't really up to it--without being some joke who's so bad she'd have to be crazy to think she could. If you actually see Stephanie as having problems that believably keep her out of the training program, the other peoples' actions make perfect sense and don't read like arbitrary dickery or author sexism.