I don't think this line of analysis works for determining that a work is sexist.

Well, I'm not sure what it means for a work as a whole to be sexist. So in so far as that doesn't seem well-defined I agree.

At least, it's not sexist in a problematic way (i.e. we need to get rid of it, or at least be aware of the sexism when reading it),

Hang on. Full stop. The idea that any form of sexism in a work means we need to get rid of that work is something I strongly, and fundamentally disagree with. No amount of sexism is a reason for censorship.

it's sexist because the world we live in is sexist and it's practically impossible to write anythong non-sexist.

Possibly, but there are degrees of sexism, and there are issues when reinforcing certain sexist norms. I'd point out that for example, Brandon Sanderson Mistborn series is an excellent example of a series without any sexism issues in how the author approach things. And that's far from the only example.

No, but neither does it do anything to adress racism, ableism, homophobia and plenty of other societal issues.

Actually, racism is definitely addressed in the context of Muggles v. wizards, and homophobia has been addressed- see the point where one of the young wizards suspects that claims about homophobia in the Muggle community are an anti-Muggle slur. That was meant as a humorous aside but it was a clear dig at certain attitudes.

I agree that Eliezer did a wonderful job of promoting Daphne and Tracey. If he hadn't the situation would look very different.

But the argument isn't just that Hermione died without accomplishing anything, but rather that it was in a context where the male wizards (Harry and Voldemort) both received substantial power boosts, where Hermione was trying to be a heroine for feminist reasons, where people had already complained about feminist issues being treated poorly in HPMoR, and then having Hermione killed without accomplishing anything specifically because the villain desired it to have an impact on the primary male protagonist. It is that totally of issues that made this so bad.