But no one can know this. Because the new rites do not spell out the proper intention of the Church (as do the old rites), then the new rites are dependent upon the intention of the minister. This is not only very novel and foreign to the history of the Church, it is impossible to determine the mind of the minister (no one is God). Cardinal Ottaviani said this is the major problem with the new mass. And this problem extends to most new rites.
No amount of investigation can solve this problem, or prove that a bishop had the proper intention...because it was all in his mind. This is so unique and dangerous as to make all new rites positively doubtful.
So, that's been the SSPX line, and as you point out, even on those grounds, it's next to impossible to know the intention. Pope Leo XIII taught that the Church can't really know it, since it belongs properly to the internal forum. I mean, even if a minister later said, "I never intended that ..." ... he COULD be lying, for whatever reason. Perhaps he later become anti-Catholic or something.
Problem is that this notion of an AMBIGUOUS Rite that then has to be supplied by the intention of the minister is 1000% a novelty. At one point in
Apostolicae Curae, Pope Leo XIII stated that any ambiguities that COULD be interpreted in a Catholic sense were nevertheless vitiated by the intention of the RITE ITSELF. Pope Leo taught that the removal of all references to the priest's power to offer sacrifice manifested an intention in the Rite to bring the Rite into conformity (or at least lack of conflict) with the errors of the Reformers (the Prots). Many Anglicans subsequently argued they were valid precisely because they did have a more Catholic view of the priesthood, and at one point they fixed the essential form, where only ambiguities remained, but Leo says that the historical origin of the Rite disambiguated any potentially-Catholic meaning into the non-Catholic one and the Rite would remain invalid in perpetuity. We see exactly that with the Novus Ordo, where the obvious reason (they openly ADMITTED this) was to take out things that might be obstacles to the "separated brethren", i.e. which contradicted their errors. Now, I believe there was an even more sinister intention ... but that's speculative, and this is what they ADMIT to doing, stripping out the same reference to the priest's power to offer sacrifice to NOT offend the Prot errors.
I hold that NO Ordinations are "absolutely null and utterly void", since you can check off every box for reasons why Pope Leo taught Anglican ones were invalid. Now, I would still in practice make it conditional, since that assessment is private judgment and could be in error, and there's no harm in using the conditional.
But the existence of Positive Doubt when the evidence from within the Rite is examined is beyond any dispute. It's only extrinsic agendas like "Well, the Pope can't promulgate an invalid Rite" ... that can be applied to "save it", except that ...
1) you're begging the question that these are valid popes
2) SSPX have long denied disciplinary infallibility
3) SSPX have also made the "non-promulgation" argument, where Montini did not legally and correctly promulgate the New Mass
So the SSPX have absolutely nothing to stand on ... with Borat's perspective being irrelevant.
See, if you claim that Disciplinary Infallibility would preclude the promulgation of invalid Rites, then the same Disciplinary Infallibility would preclude that a Pope would promulgate a BAD (even if valid) Mass that displeases God and harms souls. In that case, between that and "95% Catholic", neo-SSPX should just pack it in and formally merge with FSSP ... except that the latter might reject them as too liberal and contaminated with Modernism even for their tastes.
Oh, BTW ... the argument that the intention of the Minister can supply for an ambiguous intention in the Rite also warps the traditional definition of ministerial intention. That traditional definition has the minister intending to do WHAT the Church DOES, not intending to INTEND what the Church INTENDS. But this argument pivots intention toward intending to do what the Church intends by a Sacrament.
Let's take the Sacrament of Baptism. If you get some atheistic to poor the water and say "I baptize you in the name of the Father and of the Son and of the Holy Spirit", the intention there is absolutely ambiguous in the distilled essential form. I'm washing you (the literal meaning). Well, does that mean I'm cleaning your skin? Or is it just some symbolic act where it's really your faith that's accomplishing any effect
ex opere operantis. You just can't know from the essential formula itself (without the rest of the Rite). So then HOW ON EARTH can the intention of the atheist supply for the ambiguity there? It can't, according to the twisted SSPX perspective ... but that's not what the Church says about ministerial intention. If that atheist just knows, "well, I'm doing this thing here that Catholics do" ... that suffices for the intention. When a Novus Ordo "Bishop" is ordaining, he's CLEARLY got at least that much intention, "Yeah, I'm doing this thing that the Catholic Church does." That's it ... if the Rite itself is valid. Doesn't matter if the whole time the bishops thinks it's a bunch of hooey or even sits there in his mind thinking, "I do not intend to ordain. I do not intend to ordain." ... since he's some kind of Satanist. Even while he thinks that, making an intention contrary to what the Church intends, just by performing the Rite he's intending to DO what the Church does, and it's absolutely valid despite his contrary intention against what the Church intends.
So if the NO Rite is valid (no positive doubt), then simply by the "bishop" intending to do this "Catholic ceremony thing" ... it's valid.
This twisted duplicitous argument was concocted so that SSPX could historically justify doing conditional ordinations on a regular basis while hedging their bets against offending the Conciliars, since if they got called out they could immediately begin to equivocate by saying the Rite is valid but you never know about intention.
Now, when asked about confirmations, +Lefebvre's answer was different, where he appealed to the fact that it was not uncommon among NO "bishops" to tamper with the matter and form of Confirmation. He felt that no examination was practical or in many cases possible for "each and every case", and he felt it was not obligatory to do it, but just to administer the conditional.
Early on, in the late 1970s, +Lefebvre was asked about the episcopal rite of consecration and he said it wasn't just doubtful but invalid. Then in the early 1980s, he changed his mind ... when The Nine incident happened. The Nine were NOT all SVs, and this ordination question was one of the key points. One of the Nine reminded him about his earlier assessment of Conciliar consecrations and +Lefebvre said he had changed his mind ... "now they're valid" and pointed to ... Schmidberger (IMO suspect as infiltrator), who responded about the Eastern Rite, etc. So +Lefebvre deferred on the matter and did not study it himself. But, as we have seen with other issues aslo, in the early 1980s +Lefebvre was in a +Fellay mindeset, begging to make the "experiment of Tradition", wanting to reach a practical agreement etc. But then after he reversed and started getting more negative on NO Sacraments.
So, yes, the pervasiveness of Modernist heresy regarding the Sacraments and the nature of the Sacraments is so widespread that in probably 80% of cases the NO bishops do not intend what the Church intends. In that case, however, you could say that all NO absolutions (in Confession) are also invalid ... since it's clear that the majority of NO priests (even otherwise valid ones) don't believe in the "magical mumbo jumbo" of
ex opere operato absolution.
Conciliar Church is an absolute disaster, and at the same time it's only the needs of the faithful that give SSPX any permission and mandate to engage in their ministry or apostolate or mission, so the faithful have a right to demand certainly valid Sacraments from SSPX. Otherwise, they need to make haste back to the Conciliar Religion, since they're outside the Church and will lose their souls if they die before conversion.