As Pope Benedict XVI noted, they are in dispute over doctrine with the Church - which is why they are considered in de fact schism. Someone going to a Protestant service knows they are going to a non-Catholic service. Someone going to the SSPX are opening themselves, albeit often innocently, to doctrinal disputes which are far more subtle when propounded by a priest. No one is forbidden to attend Mass at an SSPX chapel; but the Church has ample reasons why they discourage it.
You are mistaken; they do not have ordinary jurisdiction; what has been granted to them can be withdrawn. The disputes have been going on since after the closure of Vatican 2. As there appears to be no reconciliation, and the bishops are getting older, at some point they will need more bishops. At that point I seriously doubt Rome is going to “okay” the matter given the past history, and it is more likely they will go from de facto schism to de jure schism. It is possible to have limited jurisdiction while in de facto schism, as the faculties granted are for the benefit of the laity, and nothing else.
You might want to research what Pope Benedict XVI had to say; he was crystal clear that they are in dispute over doctrinal matters, If that is not heretical, it is close enough that the differences are a matter of 'polite conversation", not fact.
And the Church says that is not the issue needed. The bishops certainly have had 40 years+*/- to work out the doctrinal matters they dispute, and have not done so. The SSPX - priests and bishops - are a minority of a minority in the Church; it is certainly not a matter that Rome has been too buy to sit down with them and lay out the Magisterial matters which the SSPX dispute.
Given the FSSP have something in the range of 35 to 39 states in which they operate, I don’t see that happening in the US - perhaps elsewhere, but they have been relatively stable here.