For me the fundamental point of Rei's character is that she is her own person, NOT Yui or even Lillith.
Not to mention that her arc and introspection scenes go into dissecting the whole environment vs. nature vs. choices thing, and that's hardly going to be ended with "Actually this has a simple answer all is inborn."
The whole beauty or crux of the whole thing is that though she has only a borrowed, faded, not-quite existence where nothing is her own there's still a person in there and she's alone and in pain, but also gets to have a few scattered moments of genuine warmth and connection amid the bleak surroundings mostly with Shinji but also a lil bit with Touji for example.
In EoE it's pretty much outright stated that she represents "the hope that people can understand each other."
People don't magically recognize ppl they share genes with; Real Life People have absolutely boinked their siblings and even parents without noticing anything out of the ordinary because they did not know they were related. Genes are not magic.
The only relation Rei has with Yui is like that of a sperm or egg donor - they might look alike and share some little quirks and basic temperament, but all this is then molded by the environment and that couldn't have been more different.
Yui's still around - she's in EVA 01. And in several episodes she makes it very clear that she wants Shinji to be piloting it while Rei is consistently the one person who doesn't guilt trip him about considering leaving (in 2.0 this is more explicit as she outright says she'll do it so he can quit and doesn't have to)
Heck even the three clones have somewhat different personalities.
Yes, she protects him, but that's not because of Genetic Mind Control (in my opinion THIS is what would show you have no grasp of her character), but at first simply because it's her job,- she has some interest because he protected her in ep 1 and because he's Gendo's kid, but that's it. To begin with she doesn't consider putting herself in danger to be such a big deal because Gendo has a dozen more of her in the basement and no one would even notice if she were swapped out (or so she thinks) ... by episode 23 she clearly DID care if Armisael absorbes her or not and told them to get out, but sacrificed herself anyways despite Misato telling her to stop because she wanted to protect Shinji & everyone else.
It's only after he rescues her from the melting plug that she cares about him in particular, once she has the pretty logical reason that he's made a point to be nice to her without any ostensible ulterior motive.
Also this tendency to stuff her in the "sister" box strikes me as something only only children could think of.
A sibling bond is a whole lot more specific than "You like this person but you don't want to boink her", They're not even on first name basis.
They could develop something like a sibling bond if they had the information and maybe a few extra years but they don're remotely come close to it in canon.
Honestly Asuka is closer to this (for all that she's still pretty far) since they live in the same household and use each other's first names and have the same parental figures (Misato & Kaji)
The closest Shinji & Rei come through this is through Gendo who actually Raised Rei and once in a while visited Shinji until three years prior, with that semi-jealously he has there sometimes before he understands the position he's in.
The commentary in the elevator scene is clearly Shinji blurting out something based on an unconscious memory of Yui that is only now occurring to him after he's spent a while piloting EVA 01.
There's not necessarily anything particularly "maternal" about Rei's towel wringing (she doesn't keep her personal space particularly clean at all; At one point Shinji tidies it up for her), Yui could have wrung out her towel a bajillion different ways and they'd still remind Shinji of her.
Then, of course, he fumbles it into something resembling a compliment and she blushes but it's more an awkward that positive reaction because she knows that the world will end in half a year and that she'll never have any kids or get married or do anything at all. It's a ridiculous embarrassing suggestion to her that she kinda likes hearing but also reminds her of what's impossible to her.
You don't necessarily have to see it as romantic (though I'd argue that it's very much hinted, I can elaborate if desired but I don't want to detract from my main point which is: ), but what is much more important, and what they certainly are (and this hill I will die on) is comrades and friends.
I mean, in ep 5 Shinji decides that he wants to get to know her 'cause they're the only two people in the awful situation of being pilots, also she's always by herself and he known what that feels like.
After this we see them consistently being nice to each other, interacting on a regular basis & worrying about each other when crap happens; See that scene in ep 22 where Asuka observes them just happily talking to each other and commenting that Shinji's already "back to usual" despite spending a month Liquefied.
As OP points out they get along, are semi-comfortable in each other's presence (since neither is super talkative this requires an actual effort - on the other hand that makes it so they actually talk instead of just being commanded around by a louder person. especially Shinji who has something of a tendency to worry what ppl think of him. )
Had the Bardiel incident not taken place when it did, I think Rei would at very least have ended up absorbed into Shinji's friend group, again she was having some interactions with Touji to the point that Hikari though he had a thing for her (Don't worry Hikki she's absolutely not his type XD) and hesitates enough to shoot at EVA 03 for Bardiel to notice & ambush her.
And of course a sad ending to the story doesn't mean that it was all meaningless; It just means that things tragically didn't work out, the reasons why it couldn't weighed out the reasons why it could have, but if the latter were zero if there would be no lost potential, no sadness, no tragedy.
Being a flawed person he doesn't do everything right here and has mistakes & shortcomings coming into play (such as when he's well-too overwhelmed with everything to try talking to her after The Big Reveal - though part of it is that he isn't sure to which extent Rei III is even the same person. ), neither goes out of their way as much as they could etc. young & inexperienced + crappy circumstances galore but that doesn't mean their friendship was completely worthless or corrupt. That happens sometimes, that people are friends. It's not unrealistic or idealistic for there to be lil bit of fleeting genuineness in the world.
tl;Dr: Wether or not they're love interests, they certainly don't have a familial or a "shallow" relationship. They're friends.
Alas poor Kaworu, I knew him well...