[Question] Are the Born probabilities really that mysterious?

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.

However, I’ve been reading Everett’s “Many Worlds” thesis, and he derives the Born probabilities (pp. 69-72) by asking what happens to an observer as the system evolves. If we posit a measure M for which the measure of a trajectory (of an observer) at one time equals the sum of the measures of each trajectory “branching” from the initial one, then he shows it must be (up to a multiplicative constant) the squared amplitude.

He then claims that this is “fully analogous” to Liouville’s theorem, which can be interpreted as a law of Conservation of Information. So taking this seriously, the Born probabilities are as inevitable as the 2nd law of thermodynamics and the uncertainty principle, among many other well-known consequences of Liouville’s theorem.

In that case, it seems to me that most of the mystery has been washed away. But I’m not quite sure to what extent he means all of that when saying “fully analogous”. In particular, I’d like to know if the Born probabilities are truly an inevitable consequence of Conservation of Information?