Quantum mechanics isn't just a modest statistical take on a classical problem. It's a complete overhaul of the mathematical and conceptual underpinnings of how we understand physics. Beyond the broad implications (destroying the pre-1900 idea that science had discovered all there was), it revolutionised chemical, atomic, nuclear, and particle physics.
It's more than just modest statistics.
It is a modest mathematical change in the underlying model which leads large behavioral changes.
It isn't classical probability theory. But isn't that far from classical probability theory really.
I disagree that you have to move to intuitionist logic, if only because I don't think you can reject the law of excluded middle outright. Either way, quantum mechanics is a mathematical treatment of atomic systems under certain premises. It involves probability, sure, but it is not itself simply a modest application of it.
There is a theorem that intuitionistic logic holds in every topos, while classical logic holds only in toposes equivalent to SET (i.e. equivalent to set theory).
You have to move to a topos which isn't equivalent to SET to formulate QM, so you effectively have to reject the law of the excluded middle, and (more importantly) redefine what you mean by negation exactly.
And, I didn't say "modest application" of probabilty theory, I said "modest change", and what I meant by that (and probably didn't say clearly) is that it is a modest weakening of part of probability theory that leads to QM (of course "modest" is in the eye of the beholder and depends on what you are looking for).
A "weakening" of a theory generally produces a more general theory.
Anyway, there is an interesting program out there involving all of this which aims to formulate probability theory and mechanics in general terms which will hold over a large range of toposes, and then say...
"Hey! if I choose do all this in the topos SET I get classical probability theory and classical mechanics! But if I do this in a topos based around the logic of vector spaces, I get QM! So maybe if I choose a topos which doesn't give classical theories or quantum theories I might get the right topos to get 'quantum' gravity!"
So 'quantum' gravity would be neither a classical nor a quantum theory.