What you put in for P and Q is irrelevant, for a simple reason. If you're appealing to Q as your only evidence for P, and Q is probably false, then you don't have good evidence for for P. If Eliezer wants to appeal to some Q as his only evidence for P, and Q is probably false, then he has failed.

Of course, if you have independent evidence for P, then you don't need to appeal to P as your evidence for Q (and you shouldn't, since P is very probably false). Here you can appeal to the independent evidence. For example, there is evidence that the earth orbits the sun that is independent of Newton's theory of gravity. It's for that reason that you find your toy examples plausible.

This doesn't work when we're talking about QM. QM is a package deal that makes predictions. Evidence for the truth of many parts of the package come from the accurate predictions the package makes.

Where there is independent evidence for the parts of the QM package Eliezer wants to appeal to, he should be appealing to those parts of the package and rely upon the independent evidence for them. Appealing to QM is just not rationally acceptable behavior for any reasonably informed persons.