(unless it was stated elsewhere, the only canonical age we have for Kaworu is 9 days)
That's the manga, which is a different canon. In the series the officially stated birthday is Second Impact, and while this may be an in-universe lie, I do think that is an important clue at what
happened at 2I. I do think in the series, the Contact Experiment with Adam that caused Adam also gave birth to Kaworu. As such, he would even be a little older than Shinji. Of course, even so he is still an angel, an Adamite - an alien, basically, so yes, he probably does tick a bit differently.
The thing is, I just don't buy the "Shinji is desperate" line of reasoning as a
complete explanation. Yes, Shinji obviously was desperate. But that only means that he latched onto Kaworu in the record time we see. Because, yes, after everything that happened here finally is a person that shows him unconditional love and affection from the beginning. But I still don't think that would even
work, or at least surely not as smoothly as it did, if at least Shinji, well, weren't inclined that way. Of course, if Shinji hadn't been in such a desperate situation, the development would have gone differently, but that's like saying "if things were different, things would be different". Either way, I do think Shinji genuinely had romantic feelings.
So I don't see why there could not be a romantic bond between the two even with a less desperate Shinji (assuming that less desperate Shinji is not already in a relationship). Sure, the development of that bond would be completely different, but I don't see why it would be impossible for it to develop at all sans the despair.
One last point: In EoE, it took the visage of Kaworu to erode Shinji's ego to the point he entered Instrumentality. GNR didn't pick Asuka or Misato. Shinji face goes from terrified to relieved and borderline adoring. The canonicity of their "relationship" in that Kaworu was closest person to Shinji at that point cannot be argued.
Ah, Instrumentality/Third Impact, where every relationship can find arguments: GNR
deliberately takes on the form of GNK in order to calm Shinji, Shinji is calmest in Rei's lap, and he loses all hope when he sees Asuka dead.
TL;DR Shinji may have a closet key somewhere, but he at least presents as straight if he's anywhere but "suicidal and willing to accept any and all affection".
"Presents as straight" seems a bit... I don't know... cynical? There are various possible reasons for why it was Kaworu acting his personal closet key (heh). The easiest it simply because so far no boy had evoked those feelings in him. I mean, just because he may be bisexual doesn't mean he has to be into
every boy. Hell, maybe he
is a hetero-leaning bisexual; there is a continuum, after all. It could be that it requires desperation for Shinji to "not present as straight" - or it could be it simply requires Kaworu