"Yes, I know you recognize Benedict XVI. So what you are saying is that SSPX is trying to get Rome to go back to the valid ordinations, consecrations and Masses which got bumped in favor of invalid ones. Is that correct?"
Unfortunately, it's not that cut and dried.
The way it looks to me, the members (by which I mean the priests and bishops) look at Catholic Tradition as it existed for, say, 1,960 years. Then they look at what happened to the Church after, say, 1965. After comparing the two, they go, "Wait a minute: something is wrong! Holy Father, please take a look at the mess that has been made!"
So, they try to get Rome to at least talk about the problems that have occurred in the wake of Vatican 2, but Rome wont do that. When it comes right down to it, the Society cannot say with absolute, 100%, magesterial authority that everything that changed (such as the ordinations, consecrations, and Masses) is wrong. What they can say is that the state of the Church has deteriorated, and that it happened after -- and most likely, as a result of -- Vatican 2. In other words, they can just point out the facts and let Catholics of good will draw their conclusions.
Again, unfortunately, Rome will not even consider the possibility that Vatican 2 was (putting it mildly) not a good idea. The Society's position is "But, look what happened after Vatican 2! Cant we at least re-address the issue?" Rome says "No."
So, that being the case, the members of the Society shake their heads, shrug their shoulders, and try to go back to the way it was before the Council because that was the last point at which the Catholic Church clearly taught and defended the Catholic Faith.
This irks the Vatican to no end, I suspect, probably because they know that the Society is not, was not, and (I pray to God) will never be in Schism. Despite all the propaganda about the Society being schismatic, clerics in the highest circles of the Church have admitted at one time or another that the reality of the situation is precisely the opposite. I'm sure that has to make them angry.
The Society does not reject a single point of traditional Catholic doctrine and maintains Catholic Tradition almost fanatically. The rest of the Church is in a shambles, yet the Society is flourishing. They see that, too.
The Society is like the 900 lb. gorilla in the room: you just cant ignore it. On the other hand, Rome pretty much ignores the sedevacantists because they will simply die out over time. They have nothing to unite them: they bicker with each other and -- most importantly -- they deny the Pope, which is totally un-Catholic. They are, in effect, protestants with a small "p". I'm sorry if that offends you, but if you think about it logically, you'll see what I mean.
Rome also has nothing to fear from those working "within the system". They "drank the Kool-Aid", so to speak. In exchange for the *privilege* of saying the traditional Latin Mass, members of the FSSP or the ICKSP had to accept the validity and veracity of Vatican 2. So, like the crazy uncle you invite to dinner every once in a while, the ordinary hierarchy tolerates the traditional (now the "extraordinary" form of the) Mass to be said, and then goes back to being Novus Ordo the rest of the time.
"What about the mission of the Church? Are they trying to do anything about that? What is the history of the active response to false ecuмenism?"
Well, the SSPX websites say it better than I ever could, but I would say that the SSPX has tried in fact to take up the mission of the Church, in that it is going forth to all peoples and trying to bring them the traditional Catholic Faith. The Society is on every continent, albeit in small numbers and far-flung locations; but they are there.
As far as the active response to false ecuмenism, I go back to my earlier statement that the Society does not reject a single point of traditional Catholic doctrine; which, by extension, means that it absolutely rejects false ecuмenism and holds only the Catholic Faith to be the True Faith.
I hope this helps.