You are arguing that PA is stronger than Turing Machines.
(There's no need for me to argue against that.)
I have been told that the answer to my question is in fact Smullyan. Cheers.
"From R.M Smullyan, Recursion Theory for Metamathematics, OUP 1993, Ch. III "Shepherdson's Theorems":
<< At the time of Goedel's proof, the only known way of showing that all r.e. sets are representable in P.A. involved the assumption of omega-consistency. Well, in 1960, A. Ehrenfeucht and S. Feferman showed that all r.e. sets are representable in every *simply* consistent axiomatizable extension of the system (R). In fact, their proof showed the following stronger result:
**Theorem E.F.** If S is any simply consistent axiomatizable Rosser system for sets in which all recursive functions of one argument are strongly definable, then all r.e. sets are representable in S.
Their proof combined a Rosser-type argument with a celebrated result of John Myhill [...]
[Shortly after, strengthenings by Putnam, Smullyan, Shepherdson...] "