The Holy Father is infallible only in certain limited circumstances . . . when making an official pronouncement ex cathedra. I believe it's only been done twice in this century.
That's quite different from Christ's promise that Peter is the rock on which He will build His church, and the gates of hell shall not prevail against it.
This writer is conflating two different issues and thus is attacking something that doesn't exist.
He's also dead wrong about the evidence from the "first several centuries" of the early Church of the primacy of Rome and its authority over the entire Church. Multiple letters survive from many bishops (I recall one from St. Irenaeus) acknowledging the primacy of the successor of Peter.
Today's Gospel tells us in John's words of Peter's primacy!