Yeah, but they're also flushed every time a sparrowhawk or a peregrine flies over, or a plane, or dog-walker, or a boat goes past, so they can clearly cope with being flushed now and again. I don't think anyone's ever proven that any bird has ever died form being flushed and prevented from feeding.
Like you I'm not aware of any proof of birds being killed by repeated flushing. If an area was repeatedly disturbed I guess the birds would leave, rather than be harassed to death.
However there was an article published about 15-20 years ago which showed that Turnstones wintering close to an English coastal resort (Margate?) were not in optimum condition when they began their spring migration. The cause of that was repeated disturbance by people (especially dog walkers).