Them doing nothing (i.e. having no effect on spacetime at all when in superposition) would be a very wild thing to happen. I can't overstate how incredibly unexpected that would be. We believe (pretty strongly) that objects with mass distort space-time when they aren't in a superposition and superposition is a continuous thing. You can be in a superposition with a very high amplitude somewhere and an arbitrarily small amplitude somewhere else. If they suddenly have no impact on space-time when in superposition something very weird happens when you continuously shift from being localised to being in superposition.

Similarly it doesn't really make sense for them to exhibit the same behaviour when in superposition of different positions to when they are not. Imagine we have an object with mass which can either be in position 1 or position 2, say the positions are a meter apart. Classically we have a description of space-time for the situation where the object is in position 1 and a different description of space-time for the situation where the object is in position 2. Now put the object in an equal superposition of positions 1 & 2. If the space-time is not "different from localised buckyball behavior" which of the two choices does it choose? What happens if it chooses one of them and we move some of the weight of the superposition to the other one?