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DanielLC comments on A case study in fooling oneself - Less Wrong
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The number of particles is discrete. As far as anyone can tell, the positions aren't. They might be if you look closer than we've managed, but that's beyond the realm of quantum mechanics.
Planck distance?
Classical physics breaks down beyond this point, but the concept of distance does not. It's not meaningful to talk about the position of something beyond this accuracy because the position is blob that's bigger than that, but the blob itself has, as far as anyone can tell, infinite resolution.
Quantum physics is built on calculus. It doesn't work in discrete systems.
Your first paragraph is reasonable, but quantum physics works in discrete systems. The most extreme case is quantum computation, which often uses finite dimensional Hilbert spaces. But older and more mainstream is lattice gauge theory, which I believe approximates QFT with a discrete quantum system.
You can approximate it with a discrete system. It's just not what quantum physics uses.
Maybe lattice gauge theory isn't what reality uses, but it's still quantum physics.