What Everett says in his thesis is that if the measure is additive between orthogonal states, it's the norm squared. Therefore we should use the norm squared of observers when deciding in how to weight their observations.

But this is a weird argument, not at all the usual sort of argument used to pin down probabilities - the archetypal probability arguments rely on things like ignorance and symmetry. Everett just says "Well, if we put a measure on observers that doesn't have weird cross-state interactions, it's the norm squared." But understanding why humans described by the Schrodinger equation wouldn't see weird cross-state probability flows still requires additional thought (that's a bit hard in the non-Hamiltonian-eigenstate observer + environment basis Everett uses for convenience).

But I think that that's an argument you can make in terms of things like ignorance and symmetry, so I do think the problem is somewhat solved. But it's not necessarily easy to understand or widespread, and the intervening decades have had more than a little muddying of the waters from all sides, from non-physicist philosophers to David Deutsch.

I don't totally understand the Liouville's theorem argument, but I think it's aimed at a more subtle point about choosing the common-sense measure for the underlying Hilbert space.

I had a very similar reaction when I first read Everett's thesis as well (after having previously read Eliezer's quantum physics sequence). I reproduced Everett's proof in “Multiple Worlds, One Universal Wave Function,” precisely because I felt like it did dissolve a lot of the problem—and I also reference a couple of other similar proofs from more recent authors in that paper which you might find interesting; see references 11 and 18.

I also once showed Everett's proof to Nate Soares (who can perhaps serve as a stand-in for Eliezer here), who had a couple of objections. IIRC, he thought that it didn't fully dissolve the problem because it a) assumed that the Born rule had to only be a function of the amplitude and b) didn't really answer the anthropic question of why we see anything like the Born rule in the first place, only the mathematical question of why, if we see any amplitude-based rule, it has to be the squared amplitude.

Epistemic status: very curious non-physicist.

Here's what I find weird about the Born rule.

Eliezer very successfully thought about intelligence by asking "how would you program a computer to be intelligent?". I would frame the Born rule using the analogous question for physics: "if you had an enormous amount of compute, how would you simulate a universe?".

Here is how I would go about it:

1. Simulate an Alternate Earth, using quantum mechanics. The simulation has discrete time. At each step in time, the state of the simulation is a wavefunction: a set of (amplitude, world) pairs. If you would have two pairs with the same world in the same time step, you combine them into one pair by adding their amplitudes together. Standard QM, except for making time discrete, which is just there to make this easier to think about and run on a computer.
2. Seed the Alternate Earth with humans, and run it for 100 years.
3. Select a world at random, from some distribution. (!)
4. Scan that world for a physicist on Alternate Earth who speaks English, and interview them.

The distribution used in step (3) determines what the physicist will tell you. For example, you could use the Born rule: pick at random from the distribution on worlds given by $P\left(amplitude\right)=|amplitud{e}^2|$. If you do, the interview will go something like this:

Simulator: Hi, it's God.

Physicist: Oh wow.

Simulator: I just have a quick question. In quantum mechanics, what's the rule for the probability that an observer finds themselves in a particular world?

Physicist: The probability is proportional to the square of the magnitude of the amplitude. Why is that, anyways?

Simulator: Awkwardly, that's what I'm trying to find out.

Physicist: ...God, why did you make a universe with so much suffering in it? My child died of bone cancer.

Simulator: Uh, gotta go.

Remember that you (the simulator) were picking at random from an astronomically large set of possible worlds. For example, in one of those worlds, photons in double slit experiments happened to always go left, and the physicists were very confused. However, by the law of large numbers, the world you pick almost certainly looks from the inside like it obeyed the Born rule.

However, the Born rule isn't the only distribution you could pick from in step 3. You could also pick from the distribution given by $P\left(\alpha \right)=|\alpha |$ (with normalization). And frankly that's more natural. In this case, you would (almost certainly, by the law of large numbers) pick a world in which the physicists thought that the Born rule said $P\left(\alpha \right)=|\alpha |$. By Everett's argument, in this world probability does not look additive between orthogonal states. I think that means that its physicists would have discovered QM a lot earlier: the non-linear effects would be a lot more obvious! But is there anything wrong with this world, that would make you as the simulator go "oops I should have picked from a different distribution"?

There's also a third reasonable distribution: ignore the amplitudes, and pick uniformly at random from among the (distinct) worlds. I don't know what this world looks like from the inside.

If you view the laws of physics as the minimal program capable of generating our observations, the Born rule is no more problematic than any other part of the laws of physics. If our universe was sampled according to a different rule, it would look completely different, just the same as if the terms in the Lagrangian were changed.

The mysterious part is not the square norm, it's that the universe looks like it conspires to present apparently non-local phenomena like the projection postulate as fully local. You can handwave it with many worlds, but it does not dissolve the mystery.

In Eliezer’s Quantum Mechanics sequence, he presents the Born probabilities as still being mysterious in our understanding. In particular, the fact that it’s the only non-linear phenomenon in quantum mechanics is considered quite strange

The non-linear phenomenon is wave function collapse, considered as an objective phenomenon.

The problem is that there is no reason to posit any additive measure and treat it as probability. It can be done, but QM itself doesn't provide any significance to the numbers you would get by squaring amplitudes.

I also think Scott Aaronson's view of this issue is interesting: https://www.scottaaronson.com/democritus/lec9.html

His opinion is that you only really have two choices for a system with inherent uncertainty. Classical probability theory, or quantum amplitudes with the Born rule. I'm not sure to what extent the argument holds up, but he makes some compelling cases that the obvious alterations to probability theory or quantum amplitudes wouldn't add up to normality.