They must specify which concrete parts of the Catholic faith, the Society of Pius X rejects. I’m not aware, but I do not think they reject anything from the deposit of faith. This is necessary to be a Catholic: to make the profession of faith as did all the saints, and they did the so-called Tridentine profession of faith, which the Society of Pius X is doing. It is the same profession of faith that the fathers of the Second Vatican Council profess. When we do this, we are Catholics, so we need not establish new items to be Catholic, as the authors you mentioned are doing. I repeat that the church has established points to be Catholic, which is in the Tridentine Vatican Profession of Faith, valid until 1967, two years after the council. Since the Society of Pius X professes this, then they are Catholics.
Now, regarding the other question about the ordination of a bishop without the approval or permission of the pope, this issue is not a concrete item in the deposit of faith. In the deposit of faith, to be an ordained Catholic bishop, he must recognize the doctrine of faith and dogma on the primacy and infallibility of the pope, and the dogmas to which the Society of Pius X is adhering. The second condition to be a Catholic bishop is to mention the pope in the Holy Mass, the most concrete expression of hierarchical communion with the pope, which has been practiced since the first millennium. For the longest time, this was the method – the majority of bishops in the ancient church were ordained without the pope's participation. However, after ordination, they expressed their communion with the pope in different ways, by sending a letter to the pope, and the pope received it and sent a reply saying that they were acknowledged, or by mentioning the pope during the mass. This was in itself already sufficient, and the church for thousands of years could not be in error with this concrete method of a bishop expressing his canonical union with the pope.
In the Middle Ages, during the second millennium, extreme papal centrism was developed, an exaggerated centrism in Rome where the Pope wanted much to control. This Roman centrism was extended after the first Vatican Council, reaching its peak today with Pope Francis.
We again have to serenely view the entire 2000 years of history, how the church lived, and how bishops expressed their union with the Pope. Thus, those arguments mentioned are invalid.
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