maybe there will be some good discrete model but so far the Plank length is not a straightforward discrete unit, not like cell in game of life.

't Hooft has been quite successful in defining QM in terms of discrete cellular automata, taking "successful" to mean that he has reproduced an impressive amount of quantum theory from such a humble foundation.

More interesting still is why reals have been so useful (and not just reals, but also complex numbers, vectors, tensors, etc. which you can build out of reals but which are algebraic objects in their own right).

This is answered quite trivially by simple analogy: second-order logics are more expressive than first-order logics, allowing us to express propositions more succinctly. And so reals and larger numeric abstractions allow some shortcuts that we wouldn't be able to get away with when modelling with less powerful abstractions.