Every game logic update, not only physics, should run on a timer that's fully independent from the frame rate.

The only place where that doesn't matter is fixed hardware - i.e. old generation consoles, before they started to make "pro" upgrades.

I think you mean timestep. The video frames get updated on one timestep (the so-called "frame rate" because it is the rate at which video frames get redrawn, the inverse of its timestep), physics gets updated on a separate timestep, and gameplay or input or network polling can be updated on its own timestep.