Do you disagree?

Yes, but then I never thought they were relatively mysterious anyhow, for the reasons you describe. They're a natural law, and that's what science is for. Neither have I ever heard any physics professors or textbooks say they're mysterious. An "explanation" of the Born probabilities would be deriving them, and some other parts of quantum mechanics, from a simpler underlying framework.

What you are saying is that truly random processes are rare under the Solomonoff prior. But it should be clear that the total mass on random processes is comparable to the total mass on deterministic processes.

"Comparable," but not the same. Qualitative estimates are not enough here.

switching from randomness to some simple law (like always output H) is unlikely for the same reason that arbitrarily changing the laws of physics is unlikely.

Nope. Changing from random to simple would reduce the size of the turing machine needed to generate the output, because a specific random string needs a lot of specification but a run of heads does not. This lowers the complexity and makes it more likely by your proposed prior. The reason that this is bad for your proposed prior and not for Solomonoff induction is because one is about your experience and one is about just the universe. So even in a multiverse where all of you "happen," thus satisfying Solomonoff induction, your prior adds this extra weighting that makes it more likely for you to observe HHHHHHHHHH.