I just bought a 1987 Kawasaki Ninja GPZ 750R that needs carb tune/rebuild attention. Get it home and first day starting it up it immediately backfired. Now the starter just buzzes, sounds like it is spinning but not engaging. Just tried taking the starter off but after removing the bolts it still is tuck in there, even though I can wiggle it around a bit. What's the most likely culprit here - broken teeth on the starter gear, the flywheel, or is it something else. And why won't it come loose?

I will try to make a suggestion, but Fixya has a motorcycle section and you are listed in the cars and trucks section.
Starters have an end part called a bendix. When the starter turns, the bendix expands on the starter shaft. This puts the gear against the flywheel. In your case, it seems the gear extended and failed to recoil back out of the way.
This is why there is no clearance now. Two different approaches to creating clearance are to remove the starter endcap and grab the stator and twist it to unscrew the extended bendix to a compact size. Or remove the sparkplugs, jack the rear wheel off the ground and engage the transmission, then rotate the rear wheel in a manner to imitate reverse.
By reversing the rotation of the wheel, it should reverse the rotation of the engine flywheel and spin the broken bendix into place. By removing the sparkplugs you eliminate an accidental ignition and you all but eliminate compression.
An extra pair of hands to wiggle the starter would help as the rear wheel is turned.
Carburetion problems can be caused by bloated gaskets. Any carb built before Ethenol was distributed will swell and block fuel ports. In Automotive, these newer gaskets for Ethenol are blue, I can not say about Motorcycle applications.
It should be possible to remove the starter either way.