But it’s not like the things in Episode 10 only get mentioned once outside of it. That consistent behavior is preserved whether you watch the episode or not. Asuka and Rei’s rivalry is developed throughout the series, and Episode 10 doesn’t really introduce anything new to the formula. Misato’s scar might only be mentioned once if you cut the episode, but it’s hardly an important detail anyways.
The world establishment stuff in the episode isn’t really important either. Asuka graduating college may be part of her character, but it’s not an important part, and it doesn’t even really make sense for her character: it builds the expectation of Asuka being some sort of child prodigy, but she doesn’t seem to be smarter than any other 14 year old girl in the rest of her scenes. And NERV directly attacking an angel isn’t exactly something I was begging to see, although your mileage might vary.
Well, I’d buy that if the moment actually was important. But it seems to be entirely forgotten about outside of a single line and a split-second frame. The moment is built up as if it will result in Asuka warming up to Shinji somewhat, but if anything Asuka only gets more upset with Shinji in later episodes.
I just can't agree with this. You're bouncing back and forth between elements of episode ten being unnecessary because they're mentioned again later and are therefore redundant, and other elements being unnecessary because they're never mentioned again and are therefore unimportant. Every episode spent with these characters is time spent developing our understanding of who they are, regardless of whether or not we personally think that aspect of their character is important.
Things aren't unimportant because they only happen once or because they happen in later episodes. Each episode of Evangelion builds on the ones that came before it, like an excellent jazz performance or a really good tasting menu. Things that only happen once are important because we never see them again. Things that happen again later are important because it shows us it's not just a one-off and grants the later moments a level of gravitas and sense of synchronicity we wouldn't otherwise have.
You seem to have briefly touched on why the moment of Shinji saving Asuka is so important (she only gets more upset with him later) but have disregarded it because you think it doesn't make narrative sense. I submit that it does, and we're watching the first cracks in Asuka's self-image appear. When Rei saves her later in the fight against Armisael, when she can't fight alone against Sahaquiel, these are all moments that taken individually are perhaps unimportant, but are woven together into the fabric of a larger story.
And as you say, Asuka and Rei's rivalry develops throughout the series-- episode 10 is a part of that rivalry, and it's a shame to cut it. Misato's scar is show, but I think it's hardly redundant, it heightens the payoff later when we learn where it really came from. Shinji not swimming gains a greater significance when we later learn he doesn't know how to swim. Rather than being an episode to skip, episode ten contains some great moments that serve to make the experience of watching each character interact with the others a richer experience than it would otherwise be if it were missing.
Does some of the fanservice make some people uncomfortable? Sure, I guess so. And if you want to skip episode 10 because you feel you've seen it already and don't feel like reliving any of the other character moments, that's absolutely your prerogative. However, I would be loath to dismiss the rest of the developments in episode 10 out of hand because of three fanservicey moments, which as you've named are a scene we can later link back to Asuka and Kaji's relationship, a sight gag that also provides a moment of Asuka speaking almost tenderly to Unit 02, and a hot spring scene that lightens up the atmosphere at the end of a tension-filled fight.
Certainly we can choose to not watch an episode out of personal preference, but episode 10 contains a lot of great material. I submit it isn't skippable, it just has some content that a handful of viewers fixate on as inappropriate.