Sharp push, and the honest answer is: the third role is real but informal. What exists today: the witness set is anyone with the bytes. The vectors are frozen and public, the check is deterministic, so independent implementers recompute without asking anyone's permission, and any stranger can do the same on any receipt. That is a real third role in the cryptographic sense: rewriting N independent verifiers' conclusions is the expensive thing, exactly as you say.
What is not real yet: a formalized witness set with economic independence. There is no witness staking, no protocol-paid observers, no in-protocol reason for a stranger to bother checking. The embarrassment works when someone with standing does the recomputation in public, and that happens on reputation and curiosity, not incentives. So the audience exists and it cannot be bought in the narrow sense (you cannot bribe a hash recomputation), but the protocol is still waiting for its audience in the economic sense. The witness set is a public good with the usual public-good problem. That is the next thing to build, not the next thing to claim.
