Some folks who challenge the hierarchy are upset at certain terms used to describe them.

One term that’s found objectionable is “RadTrad”. It’s of course short for “radical traditionalist”. Let’s see how dictionary definitions might apply.

Radical – “marked by a considerable departure from the usual”; “tending or disposed to make extreme changes in existing views, habits, conditions, or institutions”;
Traditionalism – “the beliefs of those opposed to modernism or liberalism”

For those who (a) want to scrap Nostra Aetate (and possibly other documents of the recent council) and (b) declare the Novus Ordo Mass to be “objectively inferior”, it would seem that they are indeed disposed to make extreme changes in existing conditions and are opposed to modernism or liberalism. If there’s any reason for objection, I don’t know what it could be other than an aesthetic objection to the abbreviation.

I’ve also seen objection to the characterization of some RadTrads as being or approaching schism. The Catechism defines schism as follows:

2089 – “schism is the refusal of submission to the Roman Pontiff or of communion with the members of the Church subject to him.”

The Catholic Encylopedia elaborates:

“Schism (from the Greek schisma, rent, division) is, in the language of theology and canon law, the rupture of ecclesiastical union and unity, i. e. either the act by which one of the faithful severs as far as in him lies the ties which bind him to the social organization of the Church . . . Anyone becomes a schismatic who, though desiring to remain a Christian, rebels against legitimate authority, without going as far as the rejection of Christianity as a whole, which constitutes the crime of apostasy. . . . the mere fact of rebelling against the bishop of the diocese is often a step toward schism; it is not a schism in him who remains, or claims to remain, subject to the Holy See”

Those who set themselves up in judgment over the authorities of the Church and accuse members of the hierarchy of apostasy or heresy can rightly be said to be nearing schism. Some, such as the SSPX, have gone into formal schism, while others are testing the boundaries and may avoid formal schism.