The FSSP is a compromised institution; that is the problem.
Over time, you and your family will come to get more comfortable with Vatican II. It will slowly cease to be the horrible monster THAT IT ACTUALLY IS.
You will certainly come to forget that the Church is in a horrible Crisis. After all, if you can have your Latin Mass every Sunday with the permission of Rome, what's all the fuss about? Those "Traditionalists" -- what's their problem?
The fact is that Vatican II and its aftermath has caused the greatest loss of souls since the Protestant Revolt in the 1500's -- possibly worse. I don't think we can criticize or hate Vatican II enough.
Yes, the FSSP has a valid Mass, assuming the ordinations of the priests are valid (that's not exactly a non-issue, but it's a topic for another post). I would only go there if I had *no* other options.
Jurisdiction is not an issue. The Church supplies jurisdiction for the needs of the Faithful, including the sacraments of confession and matrimony.
But many people -- mostly those who aren't deep thinkers -- forget or don't realize that ideas are powerful, and have practical results. Ideas (including doctrine) ARE important. Doctrine is of paramount importance.
Just look around at the congregation at an Indult chapel and tell me if the people aren't more liberal.
Americans especially tend to be anti-intellectual, focusing on the practical and pragmatic to the point they completely despise doctrine and the world of ideas, as if they were of no consequence.