I think two other problems with her which imo makes her an not so interesting character is first, that she does not really earn her powers and that second she uses them mostly passive.
Sora only got the keyblade because Riku fell to darkness. He had to earn the right to summon it through its journey and is still the only one right now that got it without a ritual. Him losing the blade and then getting it back through is own will and belief was a great moment. Riku is kinda the same, just that he got the blade through the ritual, lost it through is actions and then worked to regain his right to wield a keyblade and redeem himself through that. Them being where they are right now feels earned.
Kairi somehow has the PoH power. She did not work with it, she just got born with that (?). On top of that she just got her keyblade handed to her by Riku who was at that time not even a master. Of course she was still training for the battle with it and probably got the right to wield it but for me it just not felt earned. She just got it because of plot.
And she uses most of her power that she just got handed passive too. She is not actively training to use her PoH power for the battle. She probably does not even know what it can do. In all KH games where she uses it, she just believes and the power makes the stuff happen. Her hug turns Sora back (and she seemingly did not know that it would happen), her letter somehow turned up in the RoD and her belief that Sora was going to be fine somehow let him keep his form. (Bur Rikus belief in him was the first reason he even got into the final world at all - so both saved him there) The problem is that the thing that she finally trained for, battling with a keyblade, is something she utterly fails at. She kills a shadow and maybe some smaller heartless in the unseen battle and is able to defend herself against Xion for a time. Thats it. The position where she finally could have shined and be her own person and not some deus ex machina plot device, was the thing where she was the worst.
Of course a game about her could change that but they would really need a storyline where she at least starts to earn some of her powers. Not get them handed to her and we just have to accept it. She would need to be a human character with character traits and mistakes (and not writing mistakes but truly her having faults) Otherwise I will just continue seeing her as a plot device and I have a hard time seeing how I would enjoy such a game at all. Maybe a small part like the Roxas tutorial but not a whole game.
But thats just all my opinion on her.
Your main problems seem to basically boil down to you finding her less interesting because she has yet to go on a journey and show us how she earned her powers and gained mastery over them.
Her development of her powers haven't been to different from Sora and Riku's just most of it happened off-screen or has yet to happen in her character arc.
I don't see how Kairi optained the keyblade being much different from how Sora and Riku did. Kairi had to undergo (intense?) training to be able to summon the keyblade, just that once again happened off-screen. I assume Aqua, Terra and Mickey had to start out with similar training.
Riku didn't have to work for his dark powers, they were just given to him by Maleficent, so I don't see that being all that different from Kairi getting her PoH powers from birth. The difference here is Riku got a chance to horn his skill and master his powers, Kairi has been benched and hasn't gotten any guidence or chance to learn to properly harness them outside of the more passive acts she's done so far.
Considiering one of her attacks in KH3 looked like a type of Perl attack she might have started learning to harness her light powers.
I am just not sure how they would expand that. They cant just go around and change scenes so imo the only way to do that without changing canon is to give her scenes when she fights against Xion. (And maybe in the heartless battles) I am not sure if they suddenly would make her playable (because if they do that to her people would also just like to have other character playable too and I am not sure when this would fit into the current story) so it would still be imo cutscenes only. And if a short fighting scene with her and xion would really make up for the things that happen before and after that? I mean Aqua and the others are awesome fighters too and people are disappointed that they got a back seat. So I am not really sure if her character is repairable in this game and if Nomura even wants that. Heck even her JP VA was quite fine with her fighting in this game so maybe they see the disappointment but disagree with it, because its their vision on how they wanted to show her.
Yeah, I think most of it would just be added cutscenes showing her take out more heartless and look like a compitent fighter. I'd also like them to add a mob or mini boss-fight with her and Lea, to make up for their turn to partner up with Sora being cut so short.
They could also add scenes of Kairi and Axel sparing in the forest, showing us some of their training. Could be an optional scene you'd get while asking Merlin how Kairi is doing.
They might have been satified with it initially, but after this reception they also might think that the audiance didn't really get what they had intended and want to tweak it. Who knows what we'll get but I don't see it out of the realm of possibilities.
I mean, to be fair I could make the argument that 95% of anything related to the Keyblade, especially in KH III, is just "hold the key in front of you and hope really hard something will happen". Visually and even spiritually nothing changes from sealing Randall's door to using the Power of Waking to sealing a Keyhole to opening Lanes for new worlds. You point, you pray, beam of light comes out. It might open a path or kill Xemnas. We don't know.
It's not like Sora or Riku have any more insight on what they're doing beyond being able to interact with keyholes, and even Aqua or Terra who are supposedly wielders with years of scholarly teachings about the subject don't seem to be that much more sure about how Keyblades do things.
Let's not forget whatever goes with Ventus and Vanitas and their capability to forge the X-Blade.
I don't feel like faulting Kairi for being born with PoH powers, or I should also fault Xehanort for being born with outstanding talent.
We all start at different checkpoints in life and we do our best with what we have.
Now, on Sora and Riku proving their worthiness aside from "you were born right so here's a magic key" I agree with, but that's the whole crux: they can flaunt good character arcs because they had the chance to do so. Kairi was always either kidnapped or relegated on an island. Or in a training we couldn't ever see.
I was always amongst the first on saying that a Keyblade for Kairi and Lea was unnecessary and would only cheapen the whole concept of having one since both of them had their own powers and fighting style, and this feeling of mine only grew stronger after KH III and how useless their roles and Keyblades were.
So I get your points and perplexities, but at the same time I don't think Kairi is lacking so far behind other characters that she's unsalvageable. Meaning, it's not like every other character but her is a dedicated and knowledgeable scholar of their own powers and has tested their mettle at every turn.
Agreed a lot of KH is I want this and believe in it with all my heart so now it'll happen.
While I agree that Axel getting a Keyblade was unnecessary and Kairi could have joined the fight with something else, they now have keyblades and SE needs to find away to make that matter in future games. With Keyblade Transformations there's still ways to spice things up. Like Axel transforming his blade into his chakrams and occasionally throwing some sword strickes into the mix.