In 1 or 2 dimensions, random walks return to the origin infinitely often. In 3 dimensions, they have but a 34% chance. There are nontrivial qualitative differences between numbers of spatial dimensions that we don't see when we think "2? 3? 5? 179? It's just a choice of N!"

Why do you think there are only 3 or 4 or 5 or 6 or 8 or 12 or 42 or 248 or n spatial dimensions?

I think we have good reason to believe that we are in 3 spatial dimensions. But as you say:

The extra dimensions could likely not impact our system of physics in any way we can detect.

What exactly is the point of these dimensions? I see no reason to concede extra dimensions to make the fact that we are living in a simulation more probable.