The enemies and opponents of Rome confirm her primacy in the act of opposing her

In historical argument, the testimony of a hostile witness carries more weight than any friendly source. If your enemy confirms your position, the case is as strong as it can be. If your enemy confirms your position while specifically attacking it, the case is settled beyond reasonable doubt.

The history of the early Church is full of hostile witnesses to Roman primacy. Tertullian attacks the Pope as Bishop of bishops — confirming the title. The Donatists appeal to Rome while denying her jurisdiction — confirming the appeal mechanism. Cyprian disputes a Roman ruling on rebaptism — confirming that Rome makes rulings. The Eastern schismatics call Rome the first throne — confirming the rank.

None of these figures intended to strengthen the Catholic case. All of them, in the act of opposing Rome, confirmed exactly what Rome claimed about herself.