Vatican II
The Catholic Encyclopedia defines Ecumenical Councils as "those to which the bishops, and others entitled to vote, are convoked from the whole world (oikoumene) under the presidency of the pope or his legates, and the decrees of which, having received papal confirmation, bind all Christians." As such to reject an Ecumenical council is to claim an authority that cannot be possessed of a human nor even as a particular bishop.
As to this particular council, the council addresses the result of modern thinking taking so many from faith the fruition of which is seen in the modern day. The problems that resulted can be understood as a mis- or under-implementation of what the council called for. The content of the council speaks for itself and only combines contemporary resources with the very same faith that has always been. To equate the development of doctrine to "change" or "progress" is a misnomer. The Gospel, the Church, and what she offers is always what a world of sin i.e. being the instrument of salvation for a fallen world. If the solution were to become the problem, then it would cease to be what it is One, Holy, Catholic, and/or Apostolic. It makes no sense to claim that this council is illegitimate because on the one hand, you would have to claim you know what makes a council legitimate and on the other, the content of the council speaks for itself. Historically speaking, this is not the only council there have been objectors to, and this is often due to a lack of sufficent perspective.
Changes to Liturgy
The central and fundamental reasoning behind this is the belief that the Church has no authority over the form of the liturgy. This is untrue, yet it is bound to the essential nature of the liturgy i.e. its purpose, what it accomplishes, and how it accomplishes that. Whatever the form the liturgy takes it must always have a Eucharistic prayer, liturgy of the word, etc. which the Church always adheres to from the precedent of old. Altering the accidental aspects of the liturgy is well within the authority of the Church since it is a precedent not set by Christ, nor apparently by the Early Church. That actual content of Sacred Tradition is alive and well, whatever the appearance, and only the Apostles themselves i.e. the bishops together have the ordination to say aye or nay on this.
The Papacy
Using the same logic, by what information can one reliably declare that a Pope and/or succession of Popes is illegitimate? You disagree with what they say? You do not recognize them as having authority? How do you know? Indeed, there are really only two possibilities:
You are God :)
There was testimony to this fact by someone with authority, who can speak to the subject being at the election.
If this is the case, then obviously he would be often cited about this, and no such person is, it seems. The Church has had elections of multiple popes at the same time, and no one wants to repeat that occurrence. Why would it have not been dealt with?
The Nature of This Belief
This position does not differ formally from Protestantism. It claims to know the Church better than the Catholic Church, that recognizes the Pope as the authoritative figurehead of the Apostolic College. Moreover, it makes the fearful claim against Matthew 16:17-19 on which is based on the principle of Ecclesial Infallibility which is to say that God would not allow His Church to be misled. As a practical rebuttal, it also fails to note all of the theologians that study the faith, bishops much included, that do not hold this belief.
Traditionalism vs. Liberalism in Catholicism
The problem of taking either one of these as one's tribe is the fact of it being a tribe. Moreover, Catholicism is a theology of tension and balance. This means that just as much as the liturgy is meant to evoke the very beauty of Heavenly liturgy, it is equally meant to allow for the individual to communion by understanding the wording and joining oneself to the sacrifice. The Church is not a political entity though capable of being divided into parties that lobby their interest and appropriate action according to the majority. You may well find the Latin Mass more beautiful and conducive to peace and the Novus Ordo Mass not so or vice versa, it does not make one form more legitimate than the other. Claiming certain teachings as more important than others is dangerous because corrupts the real meaning of the faith the Church is not only about being welcoming but formative, it is not only about being perfect but becoming so. Perhaps your charism consists of helping people deeper into their faith, or maybe it is inviting people into the faith. This should not change your beliefs but only how they are lived. Your charism or vocation in this way is not a properly universal thing. The New Evangelization to be effective cannot afford this wanton insistence. There is humanity to Church leadership but that does not mean it is therefore without divine guidance.
Another premised judgment that is made in this matter is whether one feels things are better now or worse now than ever before, the truth is it is the same. Man is still a work in progress and the Church is still the sacrament of salvation. The trend toward giving up on Christian life began with forces outside the Church, modernism. As such the only thing novel about man's state is the psychological and philosophical devolution that no longer cares about ultimate truths and conveniently dismisses them according to those same feelings of convienience. The very idea of having a "liberal" or "traditionalist" stance on something one actually has no authority or even information to take does the very same thing just inside the Church instead of without. This infection must be so dealt with so as to only see the one faith in its truth, goodness, and beauty and to trust in the Holy Spirit and its guidance of the Church, deferring on matters that require it to avoid clinging to one's own understanding (Proverbs 3:5-6).
Divine Mercy
Using the logic mentioned above, compassion does not justify evils nor does condemning people for their actions actually witness to the mercy of God. Some feel the need to speak out about Divine Mercy, which was a series of private revelations that the now-canonized Saint Faustina received from Jesus. These private revelations are not condemned, and further, have become so popular because it is totally consistent with Church teaching. The claim that it was condemned originates from the fact Pius XII halted the dissemination of information about it in 1959 (see page 271 of this document). This is not the same thing as condemnation due to error. The next Pope can undo this action by Pius XII if it seems so appropriate and so the objection ultimately becomes a sedevacantist position.