How expensive are RNGs to implement? I'm not sure how they work. I think your idea would work though.

You might make do with a very long prime value like 4073 that just clocks around endlessly, which will rarely line up meaningfully with anything else.

It does seem memory intensive though, maybe these minor issues aren't a big deal in simple enough systems that this could ever work anyway.

Maybe you just use 127 bits or something to randomize it, probably more than enough since the goal is likely mostly to avoid aliasing. Even with multiple incoming synchronized sources if you use different mixing patterns it might work out fine.