I don’t think an API is the thing that matters here. There are quite a lot of things you can’t hack on with a client alone and actually needs server side support to function (actual moderation tooling is a prime example)
Also most APIs aren’t “designed” per se and just expose the internal representations of the projects they’re of. A “common” API would either be too “wide” enough to be unusable (hello, ActivityPub C2S) or would severely limit experimentation and innovation (good luck on building microblogs in the Lemmy API) without having so many extensions that essentially end up being a third, completely different API.