If Trial Division and Miller-Rabin are functionally equivalent (compute the same results), then they simulate each other - correct? So this is not a counterexample.

This may come down to what you mean by simulate. Certainly a computer scientist would be unlikely to describe that as a simulation. And if you are asserting that anything with the same output as something else can be considered to be simulating it then your earlier claim becomes tautological. The then relevant issue becomes that this is an example where we can "simulate" something while using much less computation than running something is in complete detail (indeed, there's very little resemblance in the internal workings).

And on another whole philosophical level, how do you know your algorithm actually computes whether the input was prime or not? (or how was that originally discovered? and did that discovery require simulation?)

I can prove things without simulating. One can look at code and determine what it does without simulating. The entire point of proving algorithms correct is that that's done by mathematical proof, not by empirical testing for small values.

If Trial Division and Miller-Rabin are functionally equivalent (compute the same results), then they simulate each other - correct?

No, not correct.