The supporting arguments conflate the absence of native recursion syntax in a model with the absence of semantic equivalence, but proven inter-reducibility of models dissolves that gap at the level of extensional computability.

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extensional computability(as used in logic and philosophy of computation): Whether something can actually be computed or calculated in practice by a machine, focusing on what outputs it produces rather than how it's described.
inter-reducibility(as used in logic and philosophy of computation): When two different systems or models can be converted into each other without losing meaning—they're equally powerful, just expressed differently.
recursion (in computing/logic)(as used in logic and computer science): When a function or process refers back to itself as part of its own definition—like a set of instructions that can call themselves to solve smaller versions of the same problem.
semantic equivalence(as used in logic and language philosophy): When two different ways of expressing something mean exactly the same thing, even if they look different on the surface.

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