Given that Unti-03 was NOT ready until its core was prepped/installed in Japan, I'd say that yes, you can change the pilots and cores, it's just horrendously wasteful to do so,
Well, I do think this has nothing to do with the main argument going on, but even so I don't think that doesn't hold. Let's look at the facts we have:
1) Asuka and Shinji can pilot their EVAs because they are ensouled by their respective mothers (and Rei can also pilot 01 because she is, in a sense, Yui's daughter - Ritsuko explicitly says in Episode 14 that Rei and Shinji can cross-synch because their profile is so similar, which I take to mean to refer to their genetic link. Likewise, Shinji can probably pilot 00 because Rei is a clone of his mother.)
2) Everyone in class 2A is a prospective pilot
3) Whenever we hear about the parents of members of class 2A they don't seem to have a mother
==>4) (Conclusion from 1-3): Nobody in class 2A has a living mother, as part of them all being prospective pilots. Also, you need this mother-child link.
5) At least in the manga, Touji's mother is dead as well.
==>6) (Conclusion from 4 and 5): The soul in EVA-03 is Touji's mother.
7) The Marduk Institute is a scam. Something else determines whether somebody (i.e. the members of class 2A) are prospective pilots.
==>8) (Conclusion from 2, 4 and 7): NERV or SEELE has the souls of those dead mothers of the class 2A members, and that is what makes them prospective pilots.
So, there are basically three variants here:
1) The soul of Touji's mother was in a core which
1a) either was sent to America for construction
1b) or installed at Matsushiro; the EVA was hence only truly finished in Japan.
2) The soul wasn't stored in a core at all, and it was in fact the soul which got transferred into the already existing, but empty, EVA core. This might be possible: We know that all EVAs had empty cores at first (EVA-01 and EVA-02 sucked their respective designers in, EVA-00 was empty until it got half of Rei). And more importantly, in Episode, uh... the one where the Reiquarium gets destroyed, Ritsuko answers the question what Rei is that she is a vessel of souls. It is entirely possible that was Rei's original purpose - transfer of souls.
None of this implies that that cores, once installed (if that is the process that happens), can get swapped or changed. OTOH, I think much speaks against it: The haphazard process of how Asuka was chosen to be a pilot, the likewise haphazard use of Shinji as replacement pilot, generally, that this all seems to have been done very, well, un-methodically. Which can be explained if NERV and SEELE simply have to work with what they have at hand as well - if Kyoko is now EVA-02, well, then they have to deal with that fact, and then Asuka has to be pilot.
@DocMatoi pointed out, the reason the EVA pilots are kids is part of Anno's point that using child soldiers IS abominable, and someone having the brainwave that you don't need to use kids would defeat the point of that part of NGE's message.
I'm not sure how you mean the second part here, but I think necessity absolutely plays a part in this: The children are basically used as tools. They are needed to defeat the angels, so everyone around them only ever uses them as tools to that effect - not just SEELE, but everyone down to their guardians and commanding officers as well. Rei and Asuka have basically been raised their whole life not as children, but just as tools for a certain effect.
I think that is part of NGE's general setup in which, well, most everyone is an asshole. And that in turn is part of NGE's general
point, that humans can't understand each other, that people will always hurt each other, that people are not nice - explained and symbolized in the show by the AT Fields. And that yet, as spelled out explicitly at the end of EoE, it might be worth living regardless of that fact.
Keep in mind what Shinji says when he restores the ability of people to have AT Fields: That he
will get hurt and betrayed again - not that there is a risk, but that it
will happen. Because that is how humans are. Everything else is self-delusion, "like a prayer". And yet, he wants to "see them all again". So, given those themes, it absolutely fits that in NGE a situation is created where everyone has an interest to use the children as tools - because people are only interested in themselves, interested in saving their own hide, and so they are using children as tools. Because that's how humans are.
If it's all just the cackling council of supervillains, that would lose its edge. Then it's easy and convenient - all just those supervillains, instead of basic human nature. Then the whole symbolism of the AT Field loses its meaning, or at least, to a degree.
Plus of course, the funny thing is, if you're right, if cores are swappable and pilots replaceable... then the kids could just walk away. Then the whole thing about Shinji's motivation loses a big part of its tension. Then he could walk away while the defence of the world is still ensured. And without Ritsuko and Gendo commenting again and again in the first three episodes how they need those children after all. And then, yeah, then he really should have walked away, given his awful treatment. Which is funny, because, it isn't that the position people always against when discussing with me?
I'm agreeing that the Children shouldn't have to pilot
But that is the point: They do have to by circumstances. They are the ones who hence save the world again and again, under great personal sacrifices, as children. And yet, everyone just takes that as granted. Misato never even thanks Shinji for it again, nevermind Asuka, never lauds them for battles, never even mention this all. Neither do the classmates. All their suffering isn't even mentioned. THAT is the problem. They don't get the proper recognition, and when I point this out and you go "but the crews!!!"... that seems very wrong-headed to me.
literal thousands of people putting in cumulative tens of thousands of hours (and in some cases years of their lives, if not effectively dying) to make it possible for the monster to be fought at all. What good is a fighter pilot if their plane has no fuel, no ammo, and hasn't had a repair since it rolled of the assembly line, and they've gotten no training, have no friendly radar, and no anything behind them?
Well, it's not like they do the pilots a
service with that, or only in as far as that the pilot is also saved from angelic impact in the end. But yes, you do need the crews for that, and I never denied that. I did say they enabled the Children to save the world again and again - but still, it's the Children who do so. Once again, the point is - while what the crews do is necessary and good, the actual heroism and sacrifice is displayed by the pilots. The crews only do regular jobs as clerks, engineers, scientists etc. They go to work in the morning, do their tasks, check out in the evening and go home again. They could also have done such a job at a lab for the scientists or,for the mechanics, a car workshop.
It isn't anything
extraordinary what those crews do. It's just having a job. Meanwhile, kids having to go out and save the world in painful life and death battles? That is extraordinary and tragic and heroic. It is something that shouldn't even exist, something that is an abomination, but has to happen for the sake of the defence of the world. And so it's the pilots who are the true heroes - the pilots who save the world, and who should be accredited with that. Saying "it was a team effort" while one side of the team merely does their job while the other side get
destroyed in teenage years by piloting seems kinda like all the "we did it as a team" stuff in school, if you know what I mean...