The Catholic Church claims to be united on doctrine, yet it seems to me there are as many divisions among Catholics as there are among Protestants. Isn’t this the case?
Not really. Division isn’t inherent in Catholicism as it is in Protestantism. To the extent there’s doctrinal disunity among Catholics, this isn’t the result of Catholics living by their own principles. It’s caused by Catholics being insufficiently Catholic–by not following the teachings of the Church.
In a sense, dissenting Catholics are really Protestants (of a sort) because, while they may not dissent from Catholic teaching on the same issues as the Reformers did, they still reject the Church’s teaching and replace it with their own ideas about what Christianity is.
Protestant disunity is due, at least in part, to Protestants following their principle of sola scriptura. Even when sin and pride are excluded from the equation, Protestants still interpreted the Bible differently on important issues–sometimes even on questions directly related to salvation (like the nature of baptism or whether Christians can lose their salvation). This points to a defective method of discerning what it is God has revealed, not merely to defective discerners.