This Protestant argument is largely a reaction to hearing a common argument coming from Catholic Apologists:
“Protestants are on shaky ground because they use private interpretation to figure things out. We, however, have the magisterium, tradition, etc.”
They object and say, “Wait, if it’s somehow bad for us to use private interpretation, why is it ok for Catholics to use it? After all, at some point, the Catholic convert had to use his own private judgment to determine that he should in fact listen to the teaching authority of the Church, and that the Holy Spirit is in fact leading him in this direction. He has to use private judgment to determine whether his current priest is a liturgical nutcase or heretic, and he has to use private judgment to determine how to apply the teachings of the Church to his own life.”
It’s a fair question for them to ask, because they really do get criticized by Catholics for their own use of private judgment.