I have experienced out of balance wheels and tires that cause the car to shake at all or just some speeds. Sometimes there are speeds or speed ranges where the vibration is not as noticeable or appears to go away. Sometimes, there are speeds or ranges of speed where the vibration appears to get much worse.
I think that rotors can contribute to or cause a shimmy problem. And, I wonder...if a tire and wheel assembly is out of balance or going out of balance, if the pulsation induced during braking may cause uneven wear on the rotor possibly aggravating the balance problem, and/or causing a separate rotor pulsation issue?
For a few years back in the seventies, some shops performed wheel balancing with the wheel and tire mounted on the car so that the entire rotating mass was balanced. That worked swell, but I have not seen that done for decades.
I had a Cutlass Ciera (14" steel wheels) that had a light shimmy issue that I never could find, but that gradually diminished over time and eventually disappeared. When I bought the car, the rotors were terrible and caused a horrible pulsation when the brakes were applied. I replaced the rotors, calipers and pads early on and balanced the wheels after, but still had a light shimmy. I wonder if the bad rotors had induced uneven wear on the axle joints that eventually smoothed out over time once the rotors were replaced, but I never did find out. I was just glad it went away.