the SSPX does not reject the authority of the Pope
They don't obey him either. All of their current bishops were consecrated without permission from Rome, and all of their current priests, aside from perhaps a few who left the real Church and defected to their organization, were similarly ordained without permission. Under canon law, they are all automatically suspended and forbidden from saying mass, but do so anyway, and these illicit ordinations continue as well.
Moreover, they have rejected the teachings of the Second Vatican Council and several Popes on religious liberty and other issues, and questioned the sacramental validity of the post-Vatican II Catholic mass.
These people are heretics and schismatics.
I suppose it'd bother me less if so many of them weren't so hateful towards so many races and religions and if they didn't hold themselves up as being better Catholics than the Pope. It is ridiculous for them to be so critical of the Eastern Orthodox and the Protestants and such for doing exactly what they did- breaking from the authority of Rome and forming their own Church. Sure, the SSPX doesn't claim to be it's own Church, but it is not in communion with Rome, and it uses it's own mass and has it's own teachings. They should just admit to being what they are- Protestants.
Nothing wrong with Protestants. I do have a problem with the SSPX, though. I think they have some deep rooted issues with denial. How can you say you are in communion with the Pope when the Pope says he isn't in communion with you? And the pure hate and evil and contempt that exists in the SSPX parishes is so grossly inappropriate that, though they may technically be called Christians, in no way do I think they reflect the true spirit of Christ.
I think they may be the least Christ-like of any Christian church or denomination I have ever heard of with the exception of the Westboro Baptist Church cultists. These guys are in second place to them when it comes to most racist hateful group of people- or at least they give the impression of being such.
Obviously, that doesn't apply to everyone who attends their chapels, but this is really not a good group to associated with, in my view. I am very open to attending masses and services from different Christian groups and take good things away from past visits to them, but I would never visit an SSPX chapel. I do not think God dwells there apart from in the literal sense that they may have a valid Eucharist.
It's just said. The old mass is a beautiful liturgy. But these people are not a historical appreciation society. They are more like a right-wing hate group with wacky conspiracy theories. I shudder to think what they must be saying about our President now. Knowing them, I would not care to read the insane racist conspiratorial ramblings that no doubt populate their message forums in regard to him and to basically everything.
Something is just not right with these folks. I think the Archbishop who founded them was struggling with some degradation of his mental faculties and possible issues with psychological health towards the end, which is sad and gives him a possible excuse. His writings and sayings were really all over the place and contradictory. And maybe some within this organization today are also less than responsible for their actions because they struggle with mental health issues. But some, I think, are just bigots and like to hate.
Pope Benedict did everything possible that he could to reach out to this people, including a tremendous gift and concession to them theologically and practically in 2007 when he opened up the old mass to all Catholic priests and said it was never abolished (Even though it indeed had been abolished) and lifted the excommunications of their bishops, and they returned his generosity and his olive branch with hate.