>turing-complete, by definition, means it can represent any software program or set of programs, including of course an operating system, compiler, etc.

Turing-complete just means that it can compute any function that a Turing machine can compute. Turing machines are universal computers, not universal machines.

My computer can compute F ⇒ a Turing machine can compute F (assuming the Church-Turing thesis).

My computer can do X ⇏ a Turing machine can do X.