What then, is the argument as to why he is not a true pope?

I think the primary argument is that he was a suspected Modernist prior to election; see also https://novusordowatch.org/john-xxiii/

I've also heard some think it suspicious he named himself after an antipope.

I think that Siri was elected and that Roncalli was uncanonically (illegitimately) installed in his place.

If Siri did not accept the papacy after being elected then he was not the Pope. The person elected has to accept the office for the election to be valid. 

If Siri did not accept the papacy after being elected then he was not the Pope. The person elected has to accept the office for the election to be valid.

The suggestion is that Siri relinquished his Papacy under duress, making Roncalli's election invalid. Remember before Roncalli's white smoke, there was a previous bout of white smoke and an announcement of an election. And then suddenly they turned back on it. 

I think that Siri was elected and that Roncalli was uncanonically (illegitimately) installed in his place.

This is is Correct :cheers:

Brought to you by the sede who has docuмented proof (pdf attached) that there have been no popes since the year 1130....

"Under pain of heresy and idolatry, a Catholic must believe there were no popes since Innocent II in 1130 and no Catholic theologians since 1250 once he sees the evidence against them and once he knows the deeper dogma that non-Catholics cannot hold offices in the Catholic Church." .....there is: Against the Siri Thesis.

Most sedevacantists believe the Vatican II popes were not true popes because they professed and taught a false religion (Vatican II) to the Universal Church.

Those are SIGNS that they are not popes, yes, but it's not an explanation for how they happened to become non-popes.  Father Cekada in particular has gone down the path of considering them heretics before their election, which conveniently renders some anti-sedevacantist arguments null and void.  He would probably be correct too, since it's unlikely that these were orthodox Catholic men who suddenly at some point during their papacy became heretics.  If they were in fact heretics, it's most likely that they were heretics from the beginning and didn't just fall into heresy after their election.

The sedevacantists argument is that the Vatican II popes were public heretics before they claimed the papacy and therefore were unable to take office under divine law.

However, Roncalli was a Cardinal in good standing up to his election.

What then, is the argument as to why he is not a true pope?

The argument is from a Church docuмent below;  The men who elected him had their own agenda, therefore it was a simoniacal election.  No need for any future declaration to declare the election void.  

Simony can be money, gift, profit or benefit etc.  
Fifth Lateran Council 1512-17 A.D.
SESSION 5
16 February 1513
[Bull renewing and confirming the Constitution against not committing the evil of simony when electing the Roman pontiff]
Inserted constitution
Julius, bishop, servant of the servants of God, for an everlasting record. From a consideration that the detestable crime of simony is forbidden by both divine and human law, particularly in spiritual matters, and that it is especially heinous and destructive for the whole church in the election of the Roman pontiff, the vicar of our lord Jesus Christ, we therefore, placed by God in charge of the government of the same universal church, despite being of little merit, desire, so far as we are able with God’s help, to take effective measures for the future with regard to the aforesaid things, as we are bound to, in accordance with the necessity of such an important matter and the greatness of the danger. With the advice and unanimous consent of our brothers, cardinals of the holy Roman church, by means of this our constitution which will have permanent validity, we establish, ordain, decree and define, by apostolic authority and the fulness of our power, that if it happens (which may God avert in his mercy and goodness towards all), after God has released us or our successors from the government of the universal church, that by the efforts of the enemy of the human race and following the urge of ambition or greed, the election of the Roman pontiff is made or effected by the person who is elected, or by one or several members of the college of cardinals, giving their votes in a manner that in any way involves simony being committed — by the gift, promise or receipt of money, goods of any sort, castles, offices, benefices, promises or obligations — by the person elected or by one or several other persons, in any manner or form whatsoever, even if the election resulted in a majority of two-thirds or in the unanimous choice of all the cardinals, or even in a spontaneous agreement on the part of all, without a scrutiny being made, then not only is this election or choice itself null, and does not bestow on the person elected or chosen in this fashion any right of either spiritual or temporal administration, but also there can be alleged and presented, against the person elected or chosen in this manner, by any one of the cardinals who has taken part in the election, the charge of simony, as a true and unquestionable heresy, so that the one elected is not regarded by anyone as the Roman pontiff.
A further consequence is that the person elected in this manner is automatically deprived, without the need of any other declaration, of his cardinal’s rank and of all other honours whatsoever as well as of cathedral churches, even metropolitan and patriarchical ones, monasteries, dignities and all other benefices and pensions of whatever kind which he was then holding by title or in commendam or otherwise; and that the elected person is to be regarded as, and is in fact, not a follower of the apostles but an apostate and, like Simon, a magicianl and a heresiarch, and perpetually debarred from each and all of the above-mentioned things. A simoniacal election of this kind is never at any time to be made valid by a subsequent enthronement or the passage of time, or even by the act of adoration or obedience of all the cardinals. It shall be lawful for each and all of the cardinals, even those who consented to the simoniacal election or promotion, even after the enthronement and adoration or obedience, as well as for all the clergy and the Roman people, together with those serving as prefects, castellans, captains and other officials at the Castel Sant’ Angelo in Rome and any other strongholds of the Roman church, notwithstanding any submission or oath or pledge given, to withdraw without penalty and at any time from obedience and loyalty to the person so elected even if he has been enthroned (while they themselves, notwithstanding this, remain fully committed to the faith of the Roman church and to obedience towards a future Roman pontiff entering office in accordance with the canons) and to avoid him as a magician, a heathen, a publican and a heresiarch. To discomfort him still further, if he uses the pretext of the election to interfere in the government of the universal church, the cardinals who wish to oppose the aforesaid election can ask for the help of the secular arm against him.