I do not know about the Society as a whole or an organization. I can only speak of the events that transpired at the chapel I attended for 2-3 years and the general attitude of the parishioners who have been attending there for much, much longer.
First came several sermons about not attending Mass at independent chapels. Since I didn't know of such a thing at the time, I just listened and shelved the information. I got a "red flag at half-mast" thought. But generally no big deal.
Then came the several sermons about not attending Masses offered by the Friars and/or indult Masses. But that was accompanied by the warning that these Masses might be questionable due to compromise with New Church. Again, no big deal.
The red flag was raised in full wind during the TWO sermons about how disobedient, hot-headed, ignorant, and child-like a certain holy priest was along with the flat out refusal to honestly or openly admit where such priest had gone and why.
Then followed several sermons regarding obedience.
Then followed the priest, himself, scouring this forum and printing out posts I had written and QUOTING THEM in sermons.
Then followed the inquisitions at the door after Mass in search of the anonymous "trouble-maker."
Not to mention the attitudes of a LARGE number of parishioners at that particular chapel who refuse to associate with those who have left primarily due to the above behavior of the priests of that chapel.
And not to mention the very recent conversation (last week) with a long time chapel goer who REGULARLY ATTENDS A NOVUS ORDO MASS as well telling me and my husband point blank that Bishop Williamson bears SOLE responsibility for causing a SCHISM in the SSPX. And this person suggesting that Bishop Williamson has been causing problems in the society for 20 years (as someone who knows him well as he served under him).
Not to mention the shear number of people who flat out refused to come to our sons first communion because the chapel was not "approved."
So, I am inclined to believe based on my experiences alone that many priests believe the SSPX IS the Catholic Church and that those who do not attend there are OUTSIDE the Church. I also believe very many chapel goers hold the same opinion.
I for one am RELIEVED that I don't have to hear the politics of what is happening inside the Society every time I go to Mass. I thank God above that I am able to have a holy priest, a good confessor, outstanding sermons, and the Sacraments freely given to ALL Catholics of good will.
As for me personally, I believe that Catholics have to find a place to go where they can receive the True Sacraments and where the True Faith can be found. I don't care how nice of a chapel you have or how many people attend there or what groups you have and how many parties you put on, without the Faith, we have nothing.