It's an interesting read. I don't think it's bad, but it's not rigorous or really aimed at anything in particular. Basically asking a discrete mathematician whether he needs continuity: no. It seems reasonable that we might need separate paradigms to think about different kinds of problem (e.g., is there a physical size of the universe vs. is there a biggest prime number) because we don't know yet if there is a theory of everything or if there are innate boundary layers.

It's a fun thinking prompt, and you can go down the rabbit hole of information theory and quantized spacetime. Like you suggest, it's perfectly fine to say "infinity does not exist" and also contemplate and operate on slice at a time.