Strawson’s full arguments are more careful and I’m not deeply familiar with them, but in the Studebaker paraphrase you quote, the central paragraph claiming to “logically refute” self-determination/free will is a basic logical fallacy (or at best, a large gap) — it’s an old paradox of causation applied to “self-determination”. As you quote, Studebaker writes:
So for self-determination to be possible, one would need to be able to self-determine the process by which one self-determines, which would require that one self-determine the process by which one self-determines the process by which one self-determines. This produces an infinite regress.
Therefore self-determination is logically impossible. Therefore […] free will is impossible.
One could just as well claim to refute determinism by the same argument:
So for determinism to be possible, some cause would need to be able to determine the process by which actions are determined, which would require that some cause determine the process by which the first cause determines the process by which actions are determined. This produces an infinite regress.
Therefore determinism is logically impossible.
The fallacy here is assuming that infinite regresses are impossible. Valid logical arguments should be well-founded, without infinite regress — but the regress Strawson is constructing here is a totally different thing. We have known since Zeno that many kinds of infinite processes and regresses can and do occur in the physical worldnature.
There can certainly particularof course be reasons why ana particular infinite regress in a particular context is contradictory, or refutedrefutes some other expectation — but that is a further argument that needs to be made. You can’t just conclude “X would lead to an infinite regress —regress; so X is impossible!”