Sort of. When generation is unable to keep up with demand, and the generators can't maintain frequency keeping, the grid can go into load-shedding mode. Progressively disconnecting consumer sections until the generators can keep up.
The segments that have hospitals will tend to be the last to go.
Right, but that doesn't sound like (an analog of) the proposal under discussion which would be "anyone, so long as they pay the uniform rate, gets the hospital treatment during brownouts".