From an information-theoretic POV, yes, you can squeeze information from less, but the definition I found is based on signal levels not transmission capacity, and it seems to have other notable consequences.

AFAICT it's around this point that you can't tell whether there is a transmission, unless you know what it looks like; tuning requires decoding and/or fancy math, not a spectrometer; communication works just fine (with proper transmission modes), but there are nontrivial practical consequences as you approach or go below the noise floor.

I wouldn't expect to see analog equipment operating below the noise floor.

Link to the proof? Or at least, does the theorem has a well-known name?

The Shannon–Hartley theorem. It assumes additive white Gaussian noise (which is a good model for most kinds of thermal-ish noise), and provides a bound on the channel capacity that practical codes closely approach.

https://en.wikipedia.org/wiki/Shannon%E2%80%93Hartley_theore...