To refresh your memory about what the One Ring is:
At this point I should point out, the analogy isn't perfect, since we're talking about a super evil ring, basically forged by the equivalent of lucifer in Tolkien's fantasy world. "Omnis comparatio claudicat nisi in puncto comparationis." Every comparison limps, except in the point of comparison.
The parts of the analogy I'm seeking to use: the fact that no one can withstand the power of the ring, when the master of the ring is wielding it. The only option is to go along with what it's making you do, or die (go into mortal sin, become of bad will, willfully reject or ignore the truth).
Even the "lesser rings" work in my analogy:
Each of the Trad groups has one (or more!) of these lesser rings, which compel men to follow them (by force of their arguments). The Ring of Papal Authority is one such ring, currently wielded by Conservative Novus Ordo. The Ring of Papal Authority and Tradition is wielded by the FSSP. And so on. The SSPX recently lost its ring, but they seek the Ring of Papal Authority and Tradition. There are lesser rings based on individuals too: the Ring of Archbishop Lefebvre. Treachery caused it to be lost to the SSPX, and now ever since then the Resistance and the SSPX are both dredging rivers as we speak to find this Ring. In the meantime, SSPX and ex-SSPX Catholics are left to their own prudence to decide which group +ABL would support today (SSPX or Resistance). So you see, the power dynamics change over time as Ring ownership changes.
Maybe there are some 3rd level rings, the least and weakest of them, which actually exist in multiple copies! So then we could have "The Ring of the Tridentine Mass" because several groups are using that Ring today to influence and bring in supporters.
But just as in LOTR, if the One Ring entered the scene, it would be all over. The One Ring contains the powers of, and controls, all the others.