I'm not in the music industry in any capacity myself, so take everything I say here with a grain of salt.

And more specifically, to address your question, I don't know enough about the recording part of music production to say whether most performers get a click-track played into their monitors or not. (I know that I've seen at least some performers using click-tracks for studio recording; but I'm not sure if that's only given if they ask for it, or if it's the studio pushing it on performer to make their jobs easier. I imagine click-tracks would always be used for multi-performer async studio recording — as otherwise you'd have performances that have conflicting BPM. But maybe not for single-performer pop/EDM songs, where it's just a vocalist laying down a track, and then everything else is done in software?)

The evidence I can cite, is from the outcome of the production process: I have a strong recollection from back in the late 2000s, of using some music library auto-tagging software that would, among other things, analyze the BPM of a song to populate the BPM ID3 field of MP3s. For at least the music I loaded into it, BPM did appear to be (mostly) quantized to nice, round numbers. (And it wasn't just the software being coarse-grained in its estimates; it did give weird numbers for unusually-produced songs.)

I do know that for any modern "produced" music (i.e. music not recorded as a live ensemble session), even if the performers did lay down their tracks at a weird BPM, the audio engineer is still going to be throwing their performances as separate tracks into a DAW. And one of the things you tell a DAW, when creating a project, is the BPM — which creates a grid of bars that tracks/samples want to snap onto. Even if you're not trimming or speed-adjusting performances so as to snap the ends of each track/sample to the DAW's grid, you're still likely snapping the beginning of each track/sample to the grid — you'd have to fight the DAW to not. Which means that, by default, if the song is "produced" enough — tracks cut up and slid around, reused, etc — then the output of this moving-and-snapping process will be a song that reads as the project's BPM.

Separately, I know that people learning to play an instrument for classical/orchestra music, often practice their instrument using a metronome (at least at the early levels.) Which perhaps gives at least some of those people the speed equivalent of perfect pitch — the ability to "lock into" certain BPMs; and to know if other performers are running fast/slow vs the "expected" BPM of the song.

I would assume that even for rock performers with no classical training, the early-level "textbook" practice for drummers and rhythm guitarists also involves playing with a metronome. It might be a lot more tenable to play a rock song at e.g. 117BPM, if everyone's just playing on the down-beat of the drums; but these performers still don't want to have a sudden, unintended shift in their BPM due to not being able to do a drum fill / complex chord change quickly enough; nor to start rushing as the song amps up, forcing everyone else into increasingly-stressful playing. In doing drills with a metronome to keep their BPM constant, these performers are likely implicitly learning to lock into certain specific BPMs as well.

My impression is that the one case where this probably isn't true, is in live jazz performance, and more specifically in live jazz improv "jam sessions." In those, the percussion (if any) isn't driving the performance, but rather is just another component of the harmony, following the (often wavering!) BPM of the lead performer.

(Is this why the classic meme of classically-trained performers not getting along with jazz artists? Because the jazz artists won't stay on beat, and this irritates some perfect-pitch-like "out of tune" feeling in the classical musician?)