Why is subtraction not part of the algebra? It’s certainly familiar to every high school math student. This omission allows the counterexample, so the reveal is a bit of a disappointment IMHO.

Subtraction is not closed over positive integers, which is untidy. The point of Tarski’s conjecture was to propose a minimal number of axioms and operations, AFAICS they define the standard semiring of positive integers (with the natural definition of exponentiation added).

(Edit: positive integers aren’t exactly a semiring because 0 is excluded, although some authors do define a semiring without the requirement of an additive identity element.)

I'm not sure, but maybe it is due to that the expression a - b can be replaced as a + (-b)?

Similarly, I think a * b and a / b can be replaced with the same trick, but then I realized it may not work on non-abelian, or where multiplicative inverse is not available...

Because subtraction is not a total operation on positive integers. Negative numbers leave the domain.

The subtraction point is interesting but I don't think it makes the result disappointing. The whole point of Tarski's problem is what follows from that very restricted set of elementary identities so finding the exact minimum countermodel under those rules still seems like a pretty satisfying result.