Because DDD was just "Kairi? Who is Kairi?" until the secret ending.
Anyways, I'm also curious about an in-universe explanation. I don't see a reason why she shouldn't take it together with Sora and Riku since, like you said, they both had to start over which means that they were all equals.
I guess Nomura still doesn't really know what he wants to do with Kairi. He wasn't sure about Xion OR Aqua and basically used them to test the waters, but I feel like it wasn't enough for him? I dunno. He doesn't seem like he ever knows what he wants Kairi to be. One time she's actually fighting, and the other game she just… disappears? She has such a minor role in Days, Re:Coded, BBS, and DDD.
I guess another reason could be they just didn't know what plots she'd have in the Disney worlds? And what she'd actually BE? Riku is Sora's Dream Eater and stuff, and I don't know what they'd do with Kairi, honestly. There also aren't any C plots that I can imagine for the Disney worlds (they could've done original stuff with Kairi but… LOL).
I wish Kairi was playable though.