Bishop Fellay of the SSPX in his March 2012 letter to the whole society gave the main impediment: “the stumbling block being the question of the present-day Magisterium…”
The SSPX apparently reject the very authority of the Council and Magisterium afterward–they reject the Council as an act of the Church’s Magisterium and claim it ushered in some different, new “magisterium” that must be completely rejected. For example, this position was expressed by Bishop Tissier during his ordination sermon at Winona on June 15, 2012 where he lays out the SSPX’s reasons for this position and concludes the Council “has no authority of teaching” and “we do not accept that the council be a true Council.”
This idea is also contained in the joint statement of the SSPX bishops. In par. 4 of their June 26, 2013 joint declaration for the 25th anniversary of their episcopal ordinations, they declare Vatican II to create a new false Magisterium distinct from the authoritative Magisterium exercised by the Church until then.
An in depth explanation of this position was written by Fr. Gleize, who, from what I understand, teaches on this subject at their Econe seminary and was heavily involved with the doctrinal discussion with Rome during Benedict’s papacy. During his written back-and-forth with Msgr. Ocariz back in 2011, he argued that the Magisterium is only really the Magisterium when it maintains perfectly the constancy of truth (and he personally sees four errors), as opposed to the Magisterium’s authenticity and authority being dependent on the unity and continuity of the subject/Church and that continuity being the ultimate guarantor of truth. It’s why he and the SSPX creates dichotomies between “eternal Rome” and the Church of the city of Rome currently on earth, between the Magisterium of the past and the living Magisterium (ie the Magisterium exercised by those currently alive earth) as different things, not one continuous subject.
Fr. Gleize summarily dismisses the teaching of Vatican II as requiring no assent at all, not even religious submission, and of begging the question in claiming to require such assent. Therefore, those four alleged errors he sees makes the Magisterium of Vatican II and recent Popes not the same one Magisterium that Our Lord has established from the Apostolic era to the end of time. It is something to be rejected completely and outright.
The actual Catholic position accounts for potential errors in non-definitive judgments and provides the proper approach for them (ie not declaring the authority completely null).
