Could the Born probabilities be basic - could there just be a basic law of physics which just says directly that to find out how likely you are to be in any quantum world, the integral over squared modulus gives you the answer? And the same law could've just as easily have said that you're likely to find yourself in a world that goes over the integral of modulus to the power 1.99999?

But then we would have 'mixed references' that mixed together three kinds of stuff - the Schrodinger Equation, a deterministic causal equation relating complex amplitudes inside a configuration space; logical validities and models; and a law which assigned fundamental-degree-of-realness a.k.a. magical-reality-fluid. Meaningful statements would talk about some mixture of physical laws over particle fields in our own universe, logical validities, and degree-of-realness.

I guess I understand better now where your dislike of the "shut up and calculate" non-interpretation of QM is coming from. You refuse to acknowledge that the Born probabilities could be a manifestation of some deeper physical law we do not yet know, and that the Schrodinger equation could be another manifestation of the same law, thus removing the need for the "third thing". The standard reaction to what I just said is "but we don't need anything else, just the Schrodinger equation", and then proceed to make extra assumptions equivalent to the Born rule, only more complicated.