I do not have a problem with it in principle; but it would imply that there are uncountably many.
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. Even if the specific branch you see around you is just part of a continuum, it must be a continuum made of parts that each have a distinct enough existence to, e.g., host an observer in a definite state. This means they can be counted (or have a cardinality), and so the only way to get a real-valued weighting is if there are continuum-many of them.
Something that I strongly suspect, but which I'm not 100% sure about, is that if branch A is supposed to be x times more likely than branch B, x not a rational number, then there must be uncountably many copies of A, and uncountably many copies of B, with the A-set being x times bigger than the B-set, according to some natural measure. The alternative would be to say that A exists once, B exists once; they're both embedded in a continuum of branches, every member of which only exists once; but the measure is non-uniform for some reason. But I think this is another version of the "A exists more than B exists" fallacy. Formally we can write down a non-uniform measure, but what it actually means is that we are counting some branches for more than others, and the only way to justify that is to suppose that the branches in question are duplicated, in proportion to the extra factor.
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. 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 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 ...
The subject has already been raised in this thread, but in a clumsy fashion. So here is a fresh new thread, where we can discuss, calmly and objectively, the pros and cons of the "Oxford" version of the Many Worlds interpretation of quantum mechanics.
This version of MWI is distinguished by two propositions. First, there is no definite number of "worlds" or "branches". They have a fuzzy, vague, approximate, definition-dependent existence. Second, the probability law of quantum mechanics (the Born rule) is to be obtained, not by counting the frequencies of events in the multiverse, but by an analysis of rational behavior in the multiverse. Normally, a prescription for rational behavior is obtained by maximizing expected utility, a quantity which is calculated by averaging "probability x utility" for each possible outcome of an action. In the Oxford school's "decision-theoretic" derivation of the Born rule, we somehow start with a ranking of actions that is deemed rational, then we "divide out" by the utilities, and obtain probabilities that were implicit in the original ranking.
I reject the two propositions. "Worlds" or "branches" can't be vague if they are to correspond to observed reality, because vagueness results from an object being dependent on observer definition, and the local portion of reality does not owe its existence to how we define anything; and the upside-down decision-theoretic derivation, if it ever works, must implicitly smuggle in the premises of probability theory in order to obtain its original rationality ranking.
Some references:
"Decoherence and Ontology: or, How I Learned to Stop Worrying and Love FAPP" by David Wallace. In this paper, Wallace says, for example, that the question "how many branches are there?" "does not... make sense", that the question "how many branches are there in which it is sunny?" is "a question which has no answer", "it is a non-question to ask how many [worlds]", etc.
"Quantum Probability from Decision Theory?" by Barnum et al. This is a rebuttal of the original argument (due to David Deutsch) that the Born rule can be justified by an analysis of multiverse rationality.