It is thus claimed either that we could not know that a prospective machine was a hypermachine after witnessing finitely many computations or that it simply would not be a hypermachine since its behaviour could be simulated by a Turing machine. Hypercomputation is thus claimed to be on shaky ground.

The former suggestion seems like the more important point here. While true that the hypercomputer's behavior can be simulated on a Turing machine, this is only true of the single answer given and received, not of the abstractly defined problem being solved. The hypercomputer still cannot be proven by a Turing machine to have knowledge of a fact that a Turing machine could not.

And so the words "shaky ground" are used loosely here. The argument doesn't refute the theoretic "existence" of recursive computation any more than it refutes the existence of theoretic hypercomputation. That finite state machines are the only realizable form of Turing machine is hardly a point in their generalized disfavor.