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This ^^ is the conclusion that I have reached as well.

I know that the SSPX doesn't do conditional baptisms in every case - but rather do their own investigation case-by-case. Where one could argue that this might not be the most imprudent course of action concerning laypeople - I think it can/could have disastrous consequences for priests coming from the N.O. So, IMO a lot of people waste ink arguing about the validly of the new rites of ordination, when at least since the 1970s anyone who is even baptized in the N.O. should be conditionally baptized and especially candidates for the traditional priesthood who were not born/raised in tradition - so as to ensure the validly of their traditional priesthood. My old SSPX priest was raised in N.O. (I don't know if he was conditionally baptized - never thought to ask). Also, I know that the SSPX priest running the Immaculata now is a Protestant convert - was he conditionally baptized? Father Albert the Dominican, was I know because he stated so online, was Bishop Williamson - anyone know? I am not trying to open a can of worms, but the Modernists are really to blame as they have sown doubt, and confusion into everything.

So, I am wondering what others think and what criteria they use to determine who to trust with making these "investigations" and determinations especially concerning seminarians for the priesthood.

The problem is that the Church *initially, always* presumes validity. As such, to indiscriminately administer the sacrament again, even conditionally *can* be a sacrilege. We have no official Church authority to tell us all NO sacraments are doubtful or invalid, hence, why the SSPX investigates as far as possible to determine invalidity or doubtful, each case individually.

Trent's Catechism....

In Conditional Baptism The Sacrament Is Not Repeated

Nor let anyone suppose that it is repeated by the Church when she baptises anyone whose previous Baptism was
doubtful, making use of this formula: If thou art baptised, I baptise thee not again but if thou art not yet
baptised, I baptise thee in the name of the Father, and of the Son, and of the Holy Ghost. In such cases Baptism
is not to be considered as impiously repeated, but as holily, yet conditionally, administered.

In this connection, however, there are some matters, in which, to the very great injury of the Sacrament, abuses
are of almost daily occurrence, and which therefore demand the diligent attention of pastors. For there are not
wanting those who think that no sin is committed if they indiscriminately administer conditional Baptism. Hence if an infant be brought to them, they think that no inquiry need be made as to whether it was previously baptised, but proceed immediately to baptise the child. Nay more, although they be well aware that the Sacrament was administered at home, they do not hesitate to repeat its administration in the Church conditionally, making use of the solemn ceremonies of the Church.

This certainly they cannot do without sacrilege and without incurring what theologians call an irregularity.
According to the authority of Pope Alexander the conditional form of Baptism is to be used only when after due
inquiry doubts are entertained as to the validity of the previous Baptism. In no other case is it ever lawful to
administer Baptism a second time, even conditionally.

The problem is that the Church *initially, always* presumes validity.

Absolutely not.  It depends on which group you come over from.  Depending on that, in many cases positive doubt is presumed.  NO (in our judgment) falls squarely into that category ... rather than, say, the Orthodox, who use very precise Sacramental Rites and among whom there's no tendency or culture of "ad libing" or deviating from those Rites.

You can read that the issue is with individuals who go around "indiscriminately" administering conditionals.  "Hey, just in case the priest got the words wrong when you were baptized as a bay, if you are not baptized, I baptize you ..."

That is the kind of scenario they're talking about, not where you have a non-Catholic religion (essentially) where the priests cannot be trusted as a rule to properly follow established Sacramental Rites.

There's nothing "indiscriminate" about administering conditional Baptism for anyone coming over from the Novus Ordo.

As per many issues in the Church today, there is error to both extremes.  Examples given are some guy who doesn't even ask whether someone was baptized and just starts rattling off the forumla "indiscriminately".  As I said, that's certainly an abuse.  So when the faithful come to your chapel, you ask them.  So, are you baptized?  If so, where and by whome were you baptized?  Then, depending on how they answer you may conditionally Baptize.  So, for instance, if they tell you, "I was baptized by the Greek Orthodox." or "I was baptized by a Maronite priest." vs. "I was 'baptized' by the Mormons [invalid]." vs. "I was baptized by some Evangelical group where we walked into the river."  Some GROUPS occasion no positive doubt (Orthodox, Eastern Catholic, etc.), whereas others inherently entail invalidity (Moron) or positive doubt (many Prot groups that just make it up as they go along).  That is what's meant by making "due inquiry", not that I need to go track down this Prot minister and interrogate him about his beliefs and take his word for it if he claims that he used the formula, etc.  If you don't make that inquiry, and start just baptizing everyone that shows up ... and it turns out they were baptized Orthodox or Maronite or some other group ... then THAT is when you commit sin by indiscriminately baptizing.  But if you ask and find it it's some Prot group, you just go ahead and do it ... conditionally.

You don't do it indiscriminately, without reasonable inquiry, but you also don't have to launch some absurd investigation to somehow prove positive doubt in every given case.