Two things to note here:
(1) the Church absolutely does claim infallibility on matters of faith and doctrine.
(2) such proclamations are only the "extraordinary mode of the Magisterium."
The full number of infallible teachings is unknowable, since it can only be developed over time. But there's no way for the Church to, for example, decide abortion is a-okay after all, even though there's been no formal ex cathedra proclamation on it, because it is a universal, historical, and quite possibly apostolic doctrine.
and traditionalist Catholics maintain what even progressives recognize as the highest-quality nonwoke education system
I think you overestimate the proportion of traditionalist Catholics. Most Catholic schools are absolutely overran by a "woke" ideology and have been for a long, long time. Traditionalist schools are only getting off the ground now and suffer from a lack of qualified educators or downright silliness as a result of that demographic, e.g. teaching creationism.
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This is the extraordinary mode of the Magisterium. The ordinary mode is the college of bishops united under the Pope throughout the world and in view of the development of doctrine coming gradually to teach the same thing "in all places and at all times." The Church absolutely does claim infallibility on the teaching of faith and morals; the Pope is guaranteed to speak infallibly when invoking the extraordinary Magisterium. Otherwise, the Pope is free to make whatever theological errors he wants provided he does not attempt to teach them to the world (e.g., if Francis were to say in a homily Christ was not God, there'd be no issue (well, except that he probably shouldn't be Pope then), but if he were to do the same in an encyclical, he'd cease to be Pope).
The actual understanding of the Church here is much, much more complex than can be serviced by a Wikipedia article or Reddit comment.
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