In particular, when you say that knowledge of particles makes something colder, makes it possible to extract work, you've gone back to the ideal observer.

I think emphatically not! To extract work, you've got to be inside the system, extracting it.

If you take the perspective of a logically omniscient perfect observer outside the system, the notion of "entropy" is pretty much meaningless, as is "probability" - you never have to use statistical thermodynamics to model anything, you just use the deterministic precise wave equation.