I'm aware of Noether's theorem and symmetries leading to conserved quantities. Excuse me if this is a straightforward result in the underlying math but as a newb to the math, I want to ask, are infinitesimals absolutely necessary for translational invariance and a corresponding conserved quantity such as momentum? Why can't a discrete or grid space-time exhibit conservation laws?
I am a newbie to the math as well, but let's take one of the conservation laws.
If a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric: from this symmetry, Noether's theorem dictates that the angular momentum of the system be conserved, as a consequence of its laws of motion. The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. It is the laws of its motion that are symmetric.
I honestly don't know if this is a valid argument against your suggestion, because I have not got the math background, but the very first line of Wikipedia's article on Noether's work says:
Noether's (first) theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.
That being the case, allied with the derivative with respect to time implied in the angular momentum example above then this, to my (very naive) understanding of the continuity of a function, would suggest a problem with modelling spacetime as a set of discrete points, rather than a differentiable continuum.