In the first place, there is the fear that the document detracts from the authority of the Second Vatican Council, one of whose essential decisions – the liturgical reform – is being called into question. Firstly, in responding to the concern that the greater provision for celebration of the Extraordinary Form would call in to question the Liturgical reforms since Vatican II:
This fear is unfounded. In this regard, it must first be said that the Missal published by Paul VI and then republished in two subsequent editions by John Paul II, obviously is and continues to be the normal Form – the Forma ordinaria – of the Eucharistic Liturgy. The last version of the Missale Romanum prior to the Council, which was published with the authority of Pope John XXIII in 1962 and used during the Council, will now be able to be used as a Forma extraordinaria of the liturgical celebration. It is not appropriate to speak of these two versions of the Roman Missal as if they were “two Rites”. Rather, it is a matter of a twofold use of one and the same rite.And secondly, Pope Benedict clearly demonstrated an expectation that it is the Missal of Paul VI, and not that of John XXIII, that should unite parish communities:
...the two Forms of the usage of the Roman Rite can be mutually enriching: new Saints and some of the new Prefaces can and should be inserted in the old Missal. The “Ecclesia Dei” Commission, in contact with various bodies devoted to the usus antiquior, will study the practical possibilities in this regard. The celebration of the Mass according to the Missal of Paul VI will be able to demonstrate, more powerfully than has been the case hitherto, the sacrality which attracts many people to the former usage. The most sure guarantee that the Missal of Paul VI can unite parish communities and be loved by them consists in its being celebrated with great reverence in harmony with the liturgical directives. This will bring out the spiritual richness and the theological depth of this Missal.Summorum Pontificum appears to me to have had two unfortunate consequences, neither of which were intended when it was promulgated. The first, which was largely transitory, was that Catholics with no attachment to the Extraordinary Form felt that they had to "take a stance", one way or another, with regard to the Extraordinary Form, when the living of a Catholic life demanded no such thing. This has largely dissipated with the passage of time (though a train of thought among Traditionalists is perhaps bringing it to the fore again). The second has been the legitimacy given to a subsequent promotion of the Extraordinary Form, more or less over and against the Ordinary Form, within the Traditionalist movement, and from within the Traditionalist movement to the wider community of the Church. The initial "headline" back in 2007-8, and maintained today, was the continued use of the term "Traditional Latin Mass", with its inherent suggestion that, juridically speaking, the Extraordinary Form was more "traditional" than the Ordinary Form, when the letter to bishops accompanying Summorum Pontificum , in speaking of two forms of the same rite, indicates that the one form is as "traditional" as the other. This has reached its ultimate destination in the recent efforts of Dr Shaw to claim the Extraordinary Form as the (only) place to find authentic Catholicism (here), something that I do not think was at all envisaged by Benedict XVI.
In summary, the Traditionalist movement has taken Summorum Pontificum as legitimising a promotion of the Extraordinary Form in a manner and a context that has no justification whatsoever in Summorum Pontificum and the accompanying letter to bishops themselves.
So what will happen if the Society of St Pius X is allowed to become a Personal Prelature and its situation with respect to the universal Church is "regularised"? Bishop Fellay's recent television interview, which gave rise to speculation about this possibility, is now online with English subtitles (my sample viewing suggests that the subtitles are very accurate to the original French); and this post, though it draws largely on a different interview, appears to me to correctly present the position of Bishop Fellay articulated in the television interview.
Bishop Fellay suggests that a number of things are already in place as far as the every day life of the Society of St Pius X is concerned that represent a degree of "regularisation" of their situation: the permission of Pope Francis that allows their priests to validly / licitly confer absolution in the Sacrament of Penance, and a certain recognition of the (strictly speaking illicit) ordination of priests by local dioceses in the place of ordination are examples. The new situation of the Extraordinary Form created by Summorum Pontificum is also relevant here, in a way that is entirely consonant with the intentions expressed by Pope Benedict in his letter to bishops. But Bishop Fellay is equally clear with regard to the Society's non-negotiables - see from about 07:20 onwards in the television interview and the paragraph "A Battle of Ideas" in this post. In summary, with regard to the teachings of the Second Vatican Council that have been controverted by the Society over the years, there is no movement on the part of the Society whatsoever. The question for Bishop Fellay and the Society is whether a suggestion that these controverted points can in some way not be considered essential as part of what is termed "Catholic" would allow them, from a "regularised" position within the Church, to continue to fight their position over and against that generally accepted in the wider Church and upheld by the conciliar and post-conciliar teaching office of the Church. Though Bishop Fellay sees some signs of this possibility being offered, he appears far from certain as to whether or not it will materialise in a full reality. Given how clear Bishop Fellay is in the interview, it is surprising to me that the speculation about a possible "regularisation" has gained as much traction in the media as it has.
Why do I find the prospect of a "regularisation" of the situation of the Society of St Pius X concerning?
Given the lack of movement of the Society over controverted issues, any "regularisation" is going to legitimise to the wider Traditionalist movement the notion that certain key teachings of the Second Vatican Council are in some way "optional" as far as being Catholic is concerned. (We are not talking here of developments after the Council that are contrary to the substance of its teaching, but of the teaching itself.) Should the Holy See be explicit in ruling this out, it appears to me unlikely that the Society will accept regularisation. Should, in the interests of charity and the promotion of communion and to avoid a rejection of the proposal by the Society, some form of "future discussion" be allowed within the process of regularisation, the precedent of the response to Summorum Pontificum and the more recent advocacy of the Extraordinary Form as the locus of authentic Catholicism, is that the Traditionalist movement will in any case conclude that the controverted issues are "optional" and seek to drive a coach and horses through the attaching conditions, to the confusion both of their own adherents and others (though I suspect that Bishop Fellay himself, on the basis of what I have seen in his television interview, has an intelligence and integrity that would not lead him to encourage such a misapprehension).
Whilst - irony of ironies - one might wish to position the controverted issues at a lower or higher place within a "hierarchy of truths" as the basis for possible future discussions between the Society and the Holy See after "regularisation", and therefore arrive at an evaluation of how central they are to being "Catholic" as a step to "regularisation", that does not make the teaching of the Council optional. But there is a nicety in this that the Traditionalist movement is unlikely to respect.