No, it doesn't, for all of the reasons given in the blog post and the subsequent discussion. The Born rule is an observable fact of nature, and you cannot derive it from the Schroedinger equation, Duetsch and Wallace's claims to the contrary notwithstanding.
Maybe you can help me understand your argument better. If the Schrodinger equation describes a density of universes, doesn't that undercut the argument in the blog post?
Edited to add my understanding: if the cardinality of the multiverse is uncountable, one has to talk about densities of branches. In that scheme, the problems which branch counting introduce do not arise.
> If the Schrodinger equation describes a density of universes,
It doesn't (see below) but even if it did...
> doesn't that undercut the argument in the blog post?
No. It's really very simple: the SE is linear. The BR is not. You can't derive a non-linear result from a linear equation.
The only thing that is even a tiny bit complicated about this is figuring out where the Deutsch-Wallace argument fails, because Deutsch and Wallace are not idiots and they hid their questionable assumption very well.
So... why does the SE not describe a "density of universes"? It's because the SE does not describe universes at all! Deutsch and Wallace specifically acknowledge this, which makes their argument all that much more untenable. The SE describes the behavior of a function called "the wave function". The wave function is not a description of anything physical. The domain of the wave function is not physical space, it is configuration space. This is the reason entanglement seems weird to us, because we live in physical space and the wave function does not.
In order to get anything physical out of the wave function -- particles, universes, anything resembling our classical reality -- you have to add some additional assumption, and any additional assumption that you add will be in some sense isomorphic to the Born rule. There is no way around this. Trying to derive the BR from the SE is like trying to prove Euclid's fifth postulate.
> The wave function is not a description of anything physical. The domain of the wave function is not physical space, it is configuration space. This is the reason entanglement seems weird to us, because we live in physical space and the wave function does not.
This is circular: you're assuming that realism about the wave function is incorrect to argue against the (tragically misnamed) many-worlds interpretation, but the only substantive claim the MWI makes is that realism about the wave function is true.
> you're assuming that realism about the wave function is incorrect
No, I'm not assuming it. I simply observe that the SR by itself does not describe the outcomes of experiments. And I'm not even talking here about quantum experiments. The SR cannot by itself even account for the fact that we live in a three-dimensional space or that there is an arrow of time. And it certainly cannot account for the Born rule, which is also an experimentally observed fact.
> the only substantive claim the MWI makes is that realism about the wave function is true.
Yes, I know. The problem with this is that it is violently at odds with observations that every human makes every moment of their lives, and it does not account for the Born rule.
> I simply observe that the SR by itself does not describe the outcomes of experiments.
No formalism describes the outcome of experiments by itself. There always needs to be an interpretation attached, and our choice of interpretation is always underdetermined by observation.
> The problem with this is that it is violently at odds with observations that every human makes every moment of their lives
Like all other interpretations of all other physical theories, MWI isn't and can't be at odds with our observations alone, because our observations alone aren't enough to imply any particular relationship between observations in general. And like all other interpretations, it is and must be at odds with competing interpretations, and in particular with the folk-physics model we're naturally inclined to use. But this means very little, because folk-physics is wrong, and this far from the paleolithic savanna, it's wrong in ways that matter. Intuition is evidence, but it's not particularly strong evidence, and it's certainly not proof.
> and it does not account for the Born rule.
Sure, it still needs a separate account of subjective probability. But so does everything else: it's just a little harder to notice in the classical regime, since the relevant probabilities are always either 1 or 0.
> No formalism describes the outcome of experiments by itself.
That is so wrong as to border on the ridiculous. The quantum mechanics formalism, i.e. the SE plus the BR together describe the outcomes of experiments. No interpretation is required. You just turn the crank on the math and predictions pop out. There is a reason that "shut up and calculate" is a thing.
The only reason people talk about interpretations at all is that the structure of the math is violently at odds with our intuitions and subjective experience, and this leaves some people feeling queasy and wanting an explanation. There's nothing wrong with that. But it has absolutely nothing to do with the power of the formalism to make predictions, which is completely independent of any interpretation. (And this is true of classical mechanics as well.)
Ultimately the MWI is saying “believe the math - configuration space is physical reality”. So you are on solid ground when you give arguments that are mathematical, however the metaphysical stuff about the self is basically talking past the MWI, which specifically does not acknowledge that as valid form of argument. I also agree with the post above that arguments about “branch counting” are bunk because you cannot count in a continuous space (see: banach-tarski and related paradoxes).
No, the MWI says to believe half of the math and to ignore the other half. The problem with that is that the second half is what allows the first half to make experimental predictions. Without the second half, there is no reason to believe the first half.
> This undercuts the critics' argument.
No, it doesn't, for all of the reasons given in the blog post and the subsequent discussion. The Born rule is an observable fact of nature, and you cannot derive it from the Schroedinger equation, Duetsch and Wallace's claims to the contrary notwithstanding.