For a brief time earlier this year, there was no pope - no bishop of Rome for the Roman Catholics. The "Camerlengo of the Holy Roman Church" (Cardinal Tarcisio Bertone held this office at that time) was in charge of Vatican City and Gandalf Castle and the few other "temporal powers" that still exist. Yet there was no "visible head" or "earthly head" or the like. There was a break, of sorts, in the supposedly "unbroken succession"! (see more on this point)
And more fundamentally, life went on. Life would have gone on had the cardinals not picked a successor. The bishop of Rome is really not necessary for anyone. People who had questions about the meaning of Scripture found answers. The Bible is no more or less clear today than it was during that time.
The Roman Catholic Church couldn't exist as such without a pope for the long run, because of various administrative tasks that fall to the pope (like appointing new bishops and elevating new cardinals). But those tasks are not tasks that are really necessary for the bishop of Rome to be doing.
What's the point? Simply that the papacy isn't necessary. It wasn't necessary in the time of the apostles and it is not necessary now. Any argument for the papacy, therefore, needs to come from some other quarter than from necessity.