Shinji and Asuka are less entirely terrible and unlikable, most of the silly school and shower scenes are cut, intro song is not as good.
The third movie is weak due to it feeling like a prequel to something else, but the first two made me unable to hate Evangelion anymore. At least not completely.
Basically if you had things you didn't like about the original series, you have a much better chance of enjoying it. If you liked the show being all "deep" (pretentious) and Shinji being a completely useless twat, you'll dislike the changes.
I didn't end up disliking either Asuka or Shinji. They certainly weren't the most pleasant people by any standards, but they were interesting and had character. I didn't really like Asuka until End of Evangelion, and then I liked her more when I re-watched the series. Conversely, I liked Shinji
until End of Evangelion, when he actually was a "useless twat."
Regarding the Biblical and Freudian concepts, I thought they made the series more interesting. Anno was a depressed guy, and the way the series is shown gives a look into his mind and the minds of his characters. The Biblical references
were thrown in to make the show mysterious (though I do not remember any actual Bible quotes appearing in the show). Few people in Japan are Christian (to my knowledge), and the cross explosions are
probably just to look pretty. But at the same time, not every cross or Biblical reference is thrown in without meaning (such as the cross that can be seen behind Kaji before his death in "He was aware that he was still a child").
Regarding "Do you love me?" and "Take Care of Yourself," I find them fascinating final looks into the characters' minds and Anno's view of the world, but I agree that
this is probably not the best way to conclude a series. Fortunately, End of Evangelion fulfilled the role of a conclusion, so the original ending can be appreciated for what it is, even if not for being a good ending.
As for what else has been said, I cannot say I recall any "tsundere harem romance." What fanservice exists is mostly too disturbing to actually be fanservice (to me, at least; plenty of people have no problem pairing Shinji with Rei). Also, I liked "Both of You, Dance Like You Want to Win!"
From what I've read here, it looks like I fall into the boat of fans(?) that enjoy a more "pretentious" Evangelion, and therefore I won't like the Rebuilds as much. I'm still going to watch them, but I really wasn't expecting to be pleased. The show doesn't really need to be remade. If the movies are too similar to the source, they seem pointless; if the movies are too different from the source, they seems unfaithful.
Anyway, that's my two-pennies-worth.