The second objection is that our world is, in fact, not continuous (with the Planck length and whatnot).

I'm not really sure where that idea came from. Quantum physics is continuous. In fact, derivatives are vital to it, and you need continuity to have them. The position of an object is spread out over a waveform instead of being at a specific spot like a billiard ball, but the waveform is a continuous function of position. The waveform has a center of mass that can be specified however much you want. Also, the Planck length seems kind of arbitrary. It means something if you have an object with size one Planck mass (about the size of a small flea), but a smaller object would have a more spread out waveform, and a larger object would have a tighter one.

The Plank length is irrelevant but quantization isn't. Specifically, with with quantum mechanics it's possible to get the ass to be in a superposition of eating from one or the other (but not in the middle) in bounded time.