I've never seen Death and Rebirth, believe it or not. Never felt much need to watch it since I got into the series long after End of Eva was out. I know the DCs added some extra scenes to make things more overt on Asuka's end.
But even in the vanilla version of the show there's a lot of development in their relationship, particularly throughout the action arc, where we see them finding an equilibrium, growing closer as comrades and saving each others' asses on the field. I think it's believable and understandable that some deeper feelings could develop out of their shared circumstances and hardships.
And that's what Shinji and Shiki are missing for me. They only live together for a short time in the movies and only go into battle together twice. Once as allies and once as enemies. Shiki is more open about having a crush on Shinji in 2.0 and starts to make a move on it but it doesn't go anywhere since she friendzones herself for Rei. It's cute to see Asuka being more open with her feelings but I find that's largely because I'm aware of the trash fire their lives became in the original. Taken on its own I don't think there's much substance behind their relationship in the movies.
That's fine with me because that particular relationship simply isn't the focus this time around, and that seems to be a deliberate choice since him failing to have her back becomes a big issue in 3.0. They made the absence a plot point. With the direction the series took, I think Asuka has graduated into Misato's old role as opposed to the one her counterpart had and Shinji's relationships are more focused around Rei, Kaworu, Misato and Gendo.
The amusing and kind of sad thing is that Shiki would probably thrive if you dropped her into NGE in Soryu's place. She doesn't have nearly as many issues and considering what she asks of her own Shinji, I think she'd have little trouble hooking up with NGE's Shinji. But it's far less interesting that way because it's more of a simple boy-meets-girl story instead of the complicated minefield they normally have.