Khomiakov pointed out that protestantism is not the negation of papism, but its universalization. Instead of one infallible man in Rome, it created a host of infallible men in London, Berlin, . . .
St. Justin Popovich expressed much the same thought, describing protestantism as the vulgarization of papism.
It's not that Protestants think of themselves as "popes", rather, they believe in the concept of the "priesthood of all believers", which Catholics believe in too when they ask the question "Who is qualified to baptise" (for example).
All Christians are enjoined to pray together and discuss the meaning of the Scriptures together. It's not like we are all a bunch of imams out there workout out our own individualized interpretation of the Scriptures and their application to real life.