I think the hate towards Evangelion is based upon one key issue.
Yes, lots of mecha shows existed before and after. But here's the big thing that makes Evangelion stand out: instead of a natural-born pilot or leader (or for that matter, the over-used "I'm going to beat you to show you the truth" bullshit that was so favored at the time), you have a child forced into piloting an alien... thing, and told to save the planet or else everyone is going to die horribly. No pressure, right?
Suddenly, we don't get the over-used "I'm a hero" lead character, and instead we see what might be the most realistic portrayal of a human being in that type of situation: he can't handle it.
Shinji is every teenage boy from a broken family. He was fucked up before, and the whole angels thing isn't helping things. Yes, he is totally a whiner and needs a good slapping/fucking from any one of the women in the show. But what the fuck do you expect? He's a goddamn child. His asshole father lied to him about everything he knows, he's been emotionally blackmailed into to piloting an alien being, the government has forced him to face horrific fights against beings that would make any normal person shit their pants, and then Shinji had to deal with it all on his own because he's too fucked up to realize he needs the support of others.
But people were (and still are) used to the archetype of the awesome guy that can do everything awesomely and overcome all the obstacles that his enemies throw at him, even if he stumbles sometimes. You see this in Naruto, Bleach, One Piece... hell, even outside of shonen manga and anime, you have the same archetype used for main characters in just about anything. Shoujo likes to have the determined young girl that manages to express her feelings for the guy; harem anime has the male eventually deciding on the girl he likes, and all the trials of the past are resolved. Think of any show you know of, and think about how the main characters eventually wins the day.
Evangelion doesn't do that. It shows what people are like facing almost certain doom daily. You see the cracks in their facades and what they do to cope with it. It's nice and comforting to see fictional characters be strong and resolute. It makes us think that we would be the same in their situation. So when we see Shinji freaking out because of all the traumatic and terrifying things he's had to deal with, we don't like it because it reminds us that we're more likely to act the same way.
It's something that most people don't like to acknowledge. We're not instinctively brave. We're hardwired to do what gives us the best chance of survival. Often, that means running, hiding or just not dealing with it.
Basically, we have a terrible double-standard for the show: anime is supposed to be about cool awesomeness. But let one specific kid in one specific series fall apart because that's what people do when they're overwhelmed beyond rationality, and that character and/or show sucks.