As you say it’s subjective, but I don’t find Shinji that annoying for most of the series. And it’s not just a downwards spiral, he and the other characters have ups and downs.

Even the conclusion of the End Of Evangelion is that something is better than nothing. A big part of Shinji’s character is hesitation of what is or isn’t worth doing, and the big final choice he’s offered when he’s at his lowest possible point is basically end the existence of humans as individuals and all the pain that goes with it or let humanity be reborn. It’s a metaphor for being on the fence about suicide, and he choses life. Which is pretty strong knowing how Hideaki Anno was himself struggles with depression when he wrote this.

I know it’s a bit stupid, but this series kinda feels like a depression toggle switch for me: It makes me feel depressed if I watch it while not depressed, but had the opposite effect if I’m already feeling depressed.