I think the problem is a matter of definition, and a conflict in common uses of the word:

1. Formally a Catholic in the eyes of the church. 2. Calls themselves a Catholic 3. Is Catholic in their beliefs.

The last has a lot of grey areas as its not clear what you need to believe. There is no formal definition. its clear you do not have to agree with the Church on every single thing. On the other hand at some point (e.g. not accepting the trinity) you are seriously at odds with Catholic beliefs.

The first two definitions might sometimes include atheists.

Do you call yourself a Googler after you quit?

You, and they, can use whatever labels serve your, or their, purpose.

But at some point it doesn't make much sense, or have much meaning, to do so.

I'm a Catholic.

> I'm a Catholic

Does it make sense to call yourself that if you fail to hold to beliefs of the Catholic Church on central issues like “Who is a Catholic”?

I mean, if we are accepting your argument that neither your belief that you are Catholic nor the Church’s beloef that you are Catholic matters and you are not Catholic despite both of those if you disagree on important matters with the teachings of the Church, what is the natural conclusion?

You are making my point.

Er, no, I am showing that your two claims conflict. That doesn't support either of them.

If one can disbelieve the Primacy of the Pope, and instead that the Pretender wearing the Pope's garments is in fact the anti-Christ, and still believe they are an adherent of Catholicism, then they have asserted a Schism in the Church and that other Catholics are Apostate. You may choose to continue to use the label "Catholic", for convenience or perhaps because you think you are the One True Catholic, but the word no longer has meaning.

For a Church to place a permanent label on a person who holds Apostate beliefs is simple paternalism. A self-declared Atheist is not a Catholic, no matter what any dude with a pallium or a ferula might have to say about it.