It is simply not accurate to say that the Society of St. Pius X outright rejected Vatican II. That is a slogan, not an analysis.
The Second Vatican Council produced 16 documents: 4 constitutions, 9 decrees, and 3 declarations. The SSPX does not say that every line of every document is false. Nor does it say that the Council was not a real council, or that nothing useful or orthodox can be found in its texts.
The SSPX position is much narrower and much more serious. It argues that certain passages in certain Vatican II documents are difficult, or even impossible, to reconcile with the Magisterium as previously elucidated by Holy Mother Church.
The principal areas of concern are religious liberty, ecumenism, and collegiality. The two most serious flashpoints are usually Dignitatis humanae, on religious liberty, and Unitatis redintegratio, on ecumenism. A third important issue arises in Lumen gentium, especially in relation to episcopal collegiality.
The problem with Dignitatis humanae is not that the SSPX believes people should be forced to believe. Catholic doctrine has never taught that faith can be coerced. The issue is whether civil society, as civil society, has duties toward the true religion, and whether religious error, as such, can properly be said to possess rights.
The problem with Unitatis redintegratio is not that the SSPX opposes charity toward non-Catholics, courteous theological discussion, or efforts to remove misunderstandings. The issue is whether modern ecumenism obscures the visible unity and uniqueness of the Catholic Church by treating separated communities as though they were partial but legitimate expressions of the Church of Christ.
The problem with collegiality is similar. The SSPX is concerned that certain formulations can blur the primacy of the Pope by speaking of the college of bishops in a way that may appear to create a parallel source of supreme authority. The very existence of the Nota Praevia shows that this was not an imaginary concern.
So the honest statement is not that the SSPX "rejects Vatican II." The honest statement is that the SSPX accepts what is traditional and orthodox in Vatican II, but resists those passages which it believes cannot be reconciled with the Magisterium as previously explained, clarified, and handed down by the Church.
That distinction matters. To say that the SSPX "rejects Vatican II" makes it sound as though the Society rejects everything contained in the Council: the Trinity, Scripture, the Incarnation, the priesthood, the episcopacy, missionary activity, Catholic education, religious life, and the universal call to holiness. That is plainly false.
The real dispute is not whether Vatican II happened. It did. The dispute is not whether every sentence of Vatican II is false. It is not. The dispute is whether certain disputed passages can truly be read in continuity with the one Magisterium of the Church, or whether they introduced ambiguity, novelty, or rupture under the language of development.