No rational argument then can be constructed to ever change a person’s mind, because we can never get to premises that people must accept.
Even if we grant this premise (must we?), the conclusion simply doesn't follow. At most, it shows that arguments won't necessarily rationally convince everyone. It remains an open possibility that some people will indeed be rationally convinced, since they may well be more committed to the truth of the premises than to the conclusion's falsity.
Still, I think there is something artificial about an argument's directionality, as demonstrated by the adage that "one man's modus ponens is another's modus tollens." Valid arguments are easily inverted, simply by switching the conclusion with a premise and negating each. Validity is preserved; the contraposed argument is logically equivalent to the original. (This refutes the common claim that logic tells us how to reason. At most, logic can provide us with wide-scope norms against inconsistency, but it cannot tell us which of the conflicting claims to give up.)
So, philosophical debate should be thought of as producing... not "arguments" per se, but logical maps -- "inconsistent triads" and the like -- to show which claims cohere best with certain others, which ones rise and fall together, etc.
It should be quite clear that this process can be rationally persuasive. We feel rational/psychological pressure to have a coherent belief set. So if someone can show that some claim P coheres better with our other beliefs than does our present belief of not-P, this may bring us to change our minds about P. And, indeed, this happens all the time.
People often hold opinions due to conceptual misunderstandings. (Think of the popular false dilemma between relativism and dogmatism.) These are easily cleared up, and any half-rational agent will change their mind upon learning of their error. Much if not most philosophy is simply a matter of overcoming sloppy thinking -- appreciating possibilities or implications that we'd previously missed.
Peter is implicitly assuming that we already have maximally coherent belief sets, so that no argument (logical map) would have any new information to give us cause to update our beliefs. But this assumption is patently false. We can - and do - learn things from others' arguments, and change our minds accordingly.
P.S. I've previously, in response to a reader's challenge, given examples of changing my mind in response to rational arguments. The issue of normativity and ultimate ends is the big one. More recently, learning about 2-D semantics radically changed my opinion of conceivability arguments. So, those are two very fundamental changes right there. (Of course, it's open to Peter to insist that the changes had non-rational causes. But that would seem unmotivated and uncharitable. I certainly think that my views have improved with time, and didn't merely "shift" in a rationally neutral fashion.)
Question for regular readers: have any arguments on this blog ever led you to revise your beliefs?