But aren't we now finding ways around the Fourier uncertainty? I believe there was a recent Nobel price for advances in microscopy.

... it's about whether QM is correct.

That one is easy: QM is not correct. It is a model - ie. a best approximation etc, but there is almost certainly going to be something, somewhere that is not entirely covered by the model. Isn't that what we all hope for: that we discover something new and amazing?

... if spacetime is quantized (there's a minimum possible distance and a minimum possible time, and all times/distances are integer multiples of these minima) then the wavefunctions wouldn't be continuous...

Ah, but continuity is a matter of topology. If space itself is quantized, the topology would have to be restricted to fit space, and wavefunctions may well be not only continuous, but also smooth, if a suitable geometry can be constructed. This, in a way, illustrates the gripes I have with QM; there is almost a culture of mysticism surrounding it (or its interpretation), that stops you from reaching a deeper understanding, because you expect it to be fundamentally impossible - so you tend to lapse back into a classical mode of consideration. Thus, the typical line of thought becomes something like "1) What would the classical scenario look like?, 2) Construct the Hamiltonian 3) Apply The Magical Transformation and get a differential equation, 4) Solve to get the wavefunction". Nowhere in this process is an understanding deeper than classical mechanics required, and that, I suspect, is why people keep talking about space being discontinuous. Einstein's genius, IMO, was that he understood that physics must be intrinsic to space, and that the resulting geometry plays a dominant role in how the laws of physics work. So, even if space turns out to be a finely minced subspace of an embedding, Cartesian space, that is not actually relevant, since all the physics - the 'reality' if you like - is confined to the geometry of that subspace, and the geometry is the only interaction there is between physical space and the embedding space.