Logically, the appeal to antiquity as a proof of authenticity would also apply to more ancient faiths, while in Fact the church of Rome is manifestly not a 2,000 year other-worldly institution, while recourse to ad hominem is another example of avoiding debate on the very thing you state, and actually interacting with the argument. In short, you simply (still) do not have one.
If you want to present the pope as the solution to divisions, then it seems you you need to either affirm unquestioning assent to any and all papal teaching, including that of councils under him, and its basis, or deal with the issue of different levels of teaching and their respective types of assent, and the problems of interpreting what ones fall under, and of the meaning of these, as well as issues in which the magisterium has not "officially" spoken on. It is those who most strongly hold to (or profess to) the authority of papal teaching that exist in schism, or sects, and reprove those who support modern popes. Cultic Catholics or cafeteria ones, though actually the term can apply to both.
And then you must deal with your manner of Roman submission and its basis, whatever that may be, and that of Scripture. Your simplistic proposal ignores reality.
And if you want to assert that the Church of Rome is a 2000 year old other worldly institution then you need to deal with the manifest absence of it and contrary aspects. That is how it works in a forum of some depth.