The Second Vatican Council is - on their official website - contrasted with authentic Catholic teaching.
Further, I quote from that same page:
Our position must be:...we refuse ...to follow the Rome of neo-Modernist and neo-Protestant tendencies which became clearly manifest during the Second Vatican Council and, after the Council, in all the reforms which issued from it. (Declaration of Archbishop Lefebvre, APPENDIX I)
I find it rather unsettling that this question is even being asked. When an injustice is done, an act of reparation must be made. The SSPX states publicly that they reject the Second Vatican Council. Hence, it is not unreasonable to ask that corresponding public statements should be made to the effect that they - the SSPX - acknowledge and accept the Second Vatican Council as an authentic expression of Catholic Tradition - not as a sort of "nyah nyah! IN UR FACE!" humiliation, but as an act of justice.
It is not sufficient amends to say: "I accept the teachings of the Catholic Church...and all things implicit therein" if, in reality, you explicitly reject the teachings of the Catholic Church as promulgated by the Second Vatican Council and all that came from it - which teaching has been handed on by five successive pontiffs and the bishops teaching in communion with them. No amends would, in fact, be made were these two positions held simultaneously.
The judgment of the Church concerning the deposit of faith takes precedence over the private judgment of an individual, or even of groups of individuals. The judgment of the Church concerning the interpretation of expressions of Tradition is given by the reigning pontiff and those bishops teaching in communion with him. The assistance of the Holy Spirit was promised by Christ in the safeguarding of this deposit (and revelation in general, in both Scripture and Tradition) and the subsequent deepening of our understanding of it and its transmission (the Indefectibility of the Church). Hence, we have it on best authority that the teaching of the Second Vatican Council is authentic, good, and true because of that totality/unity. As such, our response should be a warm reception of its teachings as being a continuation of the Tradition of the Church - or, if not a warm reception, then we are at least obligated to maintain a docile obedience and refrain from public statements that detract from said council since it comes from a legitimate Authority above whom there is no other authority.
I know that the SSPX do not consider the Second Vatican Council to be part of the Magisterium of the Church because they judge it to be a break with Tradition - but such a judgment is outside an individual's purview.
So, the insistence on the acceptance of the Second Vatican Council is motivated, I think, primarily by the sense that if you reject the legitimacy of the Second Vatican Council, then you are effectively rejecting the binding power of the Magisterium of the Church, and really and truly rejecting the Church in her totality. As we so often hear, you cannot be a faithful Catholic and pick and choose what you believe.
I'm not trying to be hard on "Traditionalists" (I consider myself a real Traditionalist in the sense that I am Catholic), nor am I trying to strike a sour note.
But one should not flippantly dismiss, however, the real and deep concerns over the implications that such an affirmation, qualified affirmation, or rejection of the Second Vatican Council holds regarding the unity and indefectibility of the Church.
The Church is one. The Church is holy. The Church is catholic. The Church is apostolic. This is the Church. This is our Faith. We are proud to profess it.
Addendum: An analogy occurs to me (and it may be faulty) - the Church's teaching against contraception has not been promulgated as an infallible dogma, ne? But it is infallible by virtue of the nature of human beings and the marital act. So, if group were to publicly *deny* this teaching and then state that they believed in the teachings of the Catholic Church and all things therein, we would be right to say "uh. Wait - you don't believe in this, and it is part of the Magisterium, so you have to recant."
Addendum II: I've noticed an equivocation going on between "an individual" and "a group." A group that makes an official statement is different from an individual in a public setting making a statement.