Rather than accepting non-determinism, this could also entail that the existence of non-deterministic classical solutions implies that such configurations don't

Rather than accepting non-determinism, this could also entail that the existence of non-deterministic classical solutions implies that such configurations don't actually exist, eg. you can't construct Norton's Dome. It turns out that that's correct, and the same with the space invader example, eg. no such thing as unbounded acceleration.