I do not have a problem with it in principle; but it would imply that there are uncountably many.

Yes, those two things seem roughly equivalent.

The reason you can't think of a superposition as just a sort of continuum without genuine parts, is that the part of reality we're observing here is objectively differentiated from what it is not.

I'm not entirely sure where you are going with this objective difference thing. The difference seems to just be that this is the part where the configurations that are us happen to be. Let's see... say the universal wave function was represented with rock's in an infinitely large desert. There are (assuming a non-obfuscated wave function representation) some rocks which, if moved, would change the part of the representation which is us. There are others which when moved wouldn't change us at all - they'd change other stuff. The universe emulator could go paint those rocks a different color if he was so inclined. That's the only 'objective' difference that I expect or require. Do you require more than that? (I sincerely do not understand what you mean by objective here and so wonder if that would satisfy you.)

Uncountably many distinct branches, each duplicated uncountably many times - at least it meets my criteria for a well-formed multiverse theory (the branches can be objectively individuated, and they have a cardinality), but it's very extravagant metaphysically.

That seemed well formed. I'm not sure that it is extravagant metaphysically. It just seems like math that could be how the universe is. The extravagance all seems to be in the stories we try to tell ourselves about the math based on our intuitions. That is, it doesn't seem like an especially complicated way for reality to be - it just seems weird to us because of the simplified models that we've been working with for convenience up till now.

I'm planning a post on forms of Many Worlds that I do think are well-defined, that will focus on approaches which I consider to be much better motivated than that one.

I'd be curious. No doubt there would be some folks complaining that lesswrongians are overstepping their bounds again into physics territory that is off limits to them but I'd enjoy reading anyhow.