For Catholics who object to Vatican II, what are the key issues?

episcopal collegiality, religious liberty

This denied the pope’s authority over the bishops. Therefore Peter’s seat, is vacant, to the extent that collegiality denied the seat authority

“ all men should be at once impelled by nature and also bound by a moral obligation to seek the truth, especially religious truth” the Fathers were effectively stating that they did not have the Monopoly of Truth, especially religious Truth!. (Declaration on Religious Freedom Dignitatis Humanae, On the Rights of The Person and of the Community.)

This was a denial that the Church has the Truth especially on religious matters, meaning they denied they were with God, who is the Truth.

“God has regard for the dignity of the human person whom He Himself created and man is to be guided by his own judgment and he is to enjoy freedom. This truth appears at its height in Christ Jesus, in whom God manifested Himself and His ways with men. (Declaration on Religious Freedom Dignitatis Humanae, On the Rights of The Person and of the Community.)

This was denial of the Redeemer. Because God rejected Man, who was purchased by the Redeemer, who is the Christ.

Only in freedom can man direct himself toward goodness. Our contemporaries make much of this freedom and pursue it eagerly; and rightly to be sure. Often however they foster it perversely as a license for doing whatever pleases them, even if it is evil. For its part, authentic freedom is an exceptional sign of the divine image within man( Vatican II Documents: Pastoral Constitution on the Church in the Modern World; Gaudium Spes: Dignity of the Human Person)

This, was a rejection of the Holy Spirit, because the Holy Spirit, is the teacher of Man, about goodness, which is the Truth of God. “ I have said all these things to you, while still with you; but the Advocate, the Holy Spirit whom the father will send in my name, will teach you all these things, and remind you of all I have said” John 14:25-26.

The Catholic Church rejects nothing that is true and holy in these religions. She regards with sincere reverence those ways of conduct and of life, those precepts and teachings which, though differing in many aspects from the ones she holds and sets forth, nonetheless often reflect a ray of that Truth which enlightens all men(Declaration on the Relationship to None Christian Religions Nostra Aetate)

This was trivialization of the Catholic faith, and equating God with idol. It was blasphemy

In fidelity to conscience, Christians are joined with the rest of men in the search for truth, and for the genuine solution to the numerous problems which arise in the life of individuals from social relationships. ( Documents of Vatican II; Pastoral Constitution on the Church in the Modern World; Gaudium Spes: The dignity of the Human Person)
This was a denial that God, is the genuine solution to our problems. Again, rejection of God.

I'd like to give you an answer on just a part of your question - I hope I understood it correctly:

Is there an ... ideology ... that describes objectors to Vatican II ...

I think nearly all religions believe that their religion was given by God and men are not allowed to change.

Many Catholics (me too) belief that Christian religion (and many of its rites) was introduced by God himself who went to earth in form of Jesus Christ.

(Note that there are also groups of Catholics who deny that.)

Followers of the SSPV and similar groups believe that the form of the catholic church and its rites did not change much in the 1900 years before Vatican II. Therefore the church before Vatican II was the religion wanted by God.

They claim that the change done during Vatican II was that dramatic that the "resulting religion" is no longer the religion wanted by God.

(My personal opinion: This claim is nonsense because the changes around the years 350-400 were MUCH more dramatic than the changes in Vatican II. And of course there were also changes during the years 400-1900.)

All the other arguments I have read up to now are either...

• ... more or less necessary consequences of this claim or ...
• ... arguments used to substantiate this claim.

The key issues

1) The Vatican II Sect masquerading as the Catholic Church is not the Catholic Church but a counterfeit church prophesied in Revelation and multiple Catholic prophecies.

2) Antipope John XXIII (the second one) and his successors are heretics and have no authority to change the Mass or anything else, since heretics are outside the Church and cannot lead that which they are not part of.

3) The Novus Ordo Mass is invalid, and mortally sinful to attend.

4) Vatican II changed the Rite of Ordination in the same way that the Anglicans changed it, rendering it invalid. Therefore confessions to Vatican II priests ordained in the new Latin rite are invalid.

5) Concerning the Jews, the Catholic Church defines them as "reprobate" because they reject Christ, the Vatican II sect teaches exactly the opposite.

6) The Catholic Church teaches outside of it no one at all is saved. The Vatican II sect teaches that you can be saved in any religion.

7) The Vatican II sect is an entirely different religion than the Catholic Church

The different groups that use the term traditionalist could refer to the FSSP, the SSPX, the SSPV, the CMRI, independent priests, conclavists, or it could refer to simply sedevacantists that don't fall in to one of these groups (of which I am one).

The SSPX is a complex topic since it is undergoing a schism right now: they are not sedevacantists, but they reject certain (or all) parts of Vatican II, notably the Novus Ordo Missae. When Vatican II happened two Archbishops (Lefebvre and Thuc) created their own lines of priests and bishops because they recognized the New Rites of Ordination and Episcopal Consecration to be invalid. The SSPX rejects the Thuc line. The Thuc line is accepted by the SSPV. The SSPX however now has 'priests' ordained in the new rite giving the traditional Mass, so the earlier distinction is rendered somewhat invalid. The CMRI and the SSPV I believe accept the Thuc (maybe only the CMRI does) line and are sedevacantists. Both groups accept the heretical NFP, BoD, and BoB. Therefore they are considered false traditionalists by people like me. The FSSP is completely in the Vatican II camp, but they offer the Latin Mass but their orders are invalid.

Well, first of all, people who object to Vatican II are an extremely small portion of Catholics. If I understand correctly, the majority of priests who use the older rites of mass (the traditional latin mass), do not form part of these groups. The groups who reject Vatican II, who form a tiny minority of Catholics, are indeed composed mostly of people attached to the group called SSPX.

If I understand correctly, they object to the Church's assertion that religious freedom is a right of people of all religions, even though the Church still holds that, while there is truth in other religions, only the Catholic religion is completely true.

This is related to a very old issue. While in the first few centuries, the Catholic Church didn't have any political power, later on, the Roman Empire made Catholicism its official religion, and after this time the idea began to develop among Catholics, that Catholic rulers should coerce their subjects to accept Christianity. This was beneficial for rulers as religion provided them with a justification and solidification of their power. And so, the mixing of politics and religion was inherited from before Christianity. After the French revolution, this alliance of "throne and altar" was threatened and the Church struggled to provide an answer to the new political conditions. Now, at Vatican II, the Church definitely rejected this political teaching, as not part of its teaching on faith and morals, judging it as incompatible with the Christian Gospel which teaches peace, and asserted religious freedom.

This has caused some disagreement with the groups you mention. Originally, only these groups like SSPX offered the older form of the mass, which caused them to grow because many people were attached to this form and the new form was thus hard to accept for them. Later, the Church allowed groups faithful to its later teachings to also celebrate the older form.

In my opinion, the old rites are very beautiful, and many new people (like me occasionally) have started attending these older rites, who never experienced them before the new rites were introduced.

People who prefer the older rites are often named traditionalists, although this term does not specify whether they accept the Second Vatican Council or not.

In addition to those mentioned in MattGutting's answer, there are others who object to Vatican II because of the damage wrought on the Church following Vatican II. Here are a few examples:

1. The council was called when there was no need to call a council. Why fix what's not broken? Hence damage has come to the Church from within the Church and that was made possible by the very people whose [primary] purpose is to safeguard the Church.
2. The ambiguity of Council Documents i.e. they are written in a way that can be interpreted in a number of ways not in line with Tradition and also in a way not contradicting Tradition. This is the way novelties have entered the Church.
3. Pope John XXIII, “Throw open the windows of the church and let the fresh air of the spirit blow through,” i.e. the rapprochement with the world, actually allowed for outside-the-church influences into the Church [e.g. from Protestantism, Freemasonry, the World, and even that the Council opened the Church to the Prince of this World.] when it should be the other way 'round, the Church acting as salt preserving the world from corruption.

Pope Emeritus Benedict XVI, aware of Vatican II being used to justify say novelties, spoke of the proper interpretation of Vatican II Council, i.e. that it ought be interpreted in light of Tradition, hence the[/his] term "hermeneutic of continuity".

Please see: What Happened To Catholicism? | Vincent Ryan Ruggiero

For changes in the Mass, i.e. how there came to be Novus Ordo, please see this article Nemesis and this NY Times Op-Ed Latin Mass Appeal. If the following is as reported, it is very sad:

How was Bugnini able to make such sweeping changes? In part because none of the popes he served were liturgists. Bugnini changed so many things that John’s successor, Paul VI, sometimes did not know the latest directives. The pope once questioned the vestments set out for him by his staff, saying they were the wrong color, only to be told he had eliminated the week-long celebration of Pentecost and could not wear the corresponding red garments for Mass. The pope’s master of ceremonies then witnessed Paul VI break down in tears.

Please note that to my understanding, this reform to the liturgy was not even called for by Vatican II.

Archbishop Annibale Bugnini, main author of the New Mass, is quoted in L'Osservatore Romano, March 19, 1965, as saying:

“We must strip from our Catholic prayers and from the Catholic liturgy everything which can be the shadow of a stumbling block for our separated brethren that is for the Protestants.”

Regarding the Church and the Jews, please see this answer to Why did the Catholic Church change so radically after Vatican II in regards to the status of Jews?

In simplest form: Vatican II declared some very important things to be true which (in the minds of those who object) are radically incompatible with previous Church tradition. As the website of Sodalitium Pianum, an offshoot of SSPX, states:

The teaching of Vatican II concerning episcopal collegiality, religious liberty, ecumenism, non-Catholics belonging to the Mystical Body of Christ, non Christian religions and particularly Judaism, relations between the Church and the modern world, is in contradiction to the teaching of the Church, of many popes, and many ecumenical Councils. The reform of the liturgy, especially of the Holy Mass, and of Canon Law, are harmful for souls, favor the Protestant heresies, and declare sometimes licit that which by divine right is in fact illicit. ... The Institute [the Institute of the Mother of Good Counsel, which sponsors Sodalitium Pianum], in front of this unprecedented crisis that involves necessarily all that have approved the conciliar documents and the subsequent reforms, realizes that it cannot accept these new doctrines contrary to faith and morals nor incite the faithful to disobedience towards the legitimate authority in the Church.

The other objection they present, again in simple terms, is that Vatican II was only a 'pastoral' council, defining no dogma [and not convened to condemn heresies or resolve a particular issue], thus lacked the doctrinal weight of past Ecumenical Councils - another reason advanced by those at odds with the post-Conciliar Church to maintain their rift with the Church. The following quotes from a Pope and a future Pope are presented to bolster their arguments e.g.:

(1)Paul VI:General audience of Jan 12,1966:"In view of the pastoral nature of the Council, it avoided any extraordinary statements of dogmas endowed with the note of infallibility, but it still provided us teaching with the authority of the ordinary Magisterium, which must be accepted with docility...." - cf. Introduction: Doctrinal Authority of Vatican II | EWTN Library

and

The Second Vatican Council has not been treated as a part of the entire living Tradition of the Church, but as an end of Tradition, a new start from zero. The truth is that this particular council defined no dogma at all, and deliberately chose to remain on a modest level, as a merely pastoral council; and yet many treat it as though it had made itself into a sort of superdogma which takes away the importance of all the rest. - cf. Cardinal Ratzinger's Remarks Regarding the Lefebvre Schism. The translated text of an address by Joseph Cardinal Ratzinger, prefect of the Congregation for the Doctrine of the Faith, given July 13, 1988, in Santiago, Chile before that nation's bishops.

Since the Church does not change her teaching, traditionalists who hold beliefs of this sort feel they're faced with the choice to reject all of Church tradition as they understand it, or to reject Vatican II.