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Another consideration regarding Trent is around the negative formula, "justification cannot happen without Baptism or the desire for it".  Even if you read it the BoDer way, this falls short of actually teaching that justification CAN happen with the votum alone.

To say that ... you can't be justified without at least having the desire for it ... this just means that Catholics must hold that the Sacrament of Baptism is necessary AT LEAST in desire.  ByzCat, not a Feeneyite, agreed with this assessment.  To say that must believe something AT THE MINIMUM does not equate to positively teaching that votum suffices.

There's no Canon in Trent which says that you must believe that the votum alone suffices for justification.  Trent states that if anyone believes that the Sacraments are not necessary for salvation at least by desire, let him be anathema.  All this says is that you're absolved of heresy if you hold to BoD.  This leaves the question open and allows the BoD position, but it does not actively teach it.

Again, the phrase "justification cannot happen without" is very curious.  This speaks to necessary but but not sufficient cause.  In the case of Confession, Trent teaches ACTIVELY that sin can be remitted by the Sacrament and/or the intention (firm resolution) to receive it. Trent uses "at least" and an inclusive "either...or".  But in the case of Baptism, Trent uses the passive voice and does not throw in these terms ... when it easily could have done both.  So in the case of Confession, Trent teaches that perfect contrition along with at least the resolution to go to Confession does suffices for justification.  In the case of Baptism, even if you read this the BoDer way, Trent teaches that it is necessary for justification ("cannot happen without") but falls short of teaching it suffices on its own.

Wow..I just posted something and it just never posted!  Let's try it again:

Can I just say I love the math terms!?  I love Math.  It is so logical and universal.

I think the bolded is the critical point [and as you say the true "core concept"].  When my Jєωιѕн father died 5 years ago, I had traditional Catholics [not Novus Ordo] say to me, "Well, you never know.  He may have died with BOD".  Even then, in my grief, I thought how odd. 

So, on a practical level, this means that a Catholic can never know whether a non-Catholic [known or not known] was saved or not.  If that is the case, it calls into question, why the need for Sacramental Baptism in the first place?

I won't deny it completely because I see it in the later catechisms, but something is off.

Indeed, something is off with the whole thing.  Part of what's off is that the notion of BoD was gradually extended from its original context, asking whether a catechumen who died before Baptism could be saved.  That was the big motivation that prompted the speculation in the first place, where people saw seemingly-good and seemingly-devout catechumens pass away before Baptism.  St. Augustine actually explains that.  People were disturbed by the idea that (to paraphrase St. Augustine) devout catechumens occasionally died before Baptism, but some scoundrels who postponed Baptism til the end so they could continue on in sin.  But St. Augustine rejected that reasoning and said one should not think in those terms "if one wishes to be Catholic" and that it leads to a "vortex of confusion".  This was in his later years when he changed his mind about BoD.  Some of the strongest anti-BoD statements in existence come from St. Augustine ... and yet the subsequent BoD theorists ... St. Bernard (who was largely responsible for the resurgence of the idea after it had faded away for 600 years) and Pope Innocent ... they both based their opinion on the "authority of Augustine".  But St. Augustine never taught the opinion with authority.  He said, "Having considered the matter over and over again [gone back and forth about it], I FIND that [BoD theory]."  Notice, "I find", clearly indicating that it's speculative authority.  Had this been the consistent teaching of the Church, St. Augustine would have certainly said so, would not have gone back and forth on it, and would not have said "I find," but would have taught it with authority.  So this tentative speculative statement is what the later BoDers referred to as the "authority" of St. Augustine?

Here's a great (albeit lengthy writeup):
https://catholicism.org/baptism-of-desire-its-origin-and-abandonment-in-the-thought-of-saint-augustine.html

I must commend your great faith, where given that your father who passed away was Jєωιѕн, you nevertheless have not had a knee-jerk emotional reaction to rally behind the notion of BoD.  I believe that many of the most ardent proponents of BoD are motivated by emotional considerations and not by faith and reason.  One of them recently said something to the effect that "I refuse to believe that all those millions of souls in the New World before it was discovered had been lost."  And this emotional reaction to the New World discovery was what prompted the invention of the novel "Rewarder God" theory, which when when combined with an extended "implicit" view of Baptism of Desire, could allow them to speculate that all those could have been saved.  There's no actual theological proof for Rewarder God theory, and it runs counter to the unanimous teaching and belief of the Church for the first 1500+ years.

You're right about it being tied to emotion. It's a hard thought to realize that my Lutheran grandpa is in hell, given that he died suddenly; but that's the grave reality of the situation here. A reality which has been effectively lost on so many "Catholics" these days who have grown lukewarm and complacent with the "nice" sentiment that non-Catholics are saved. It's precisely why almost no one evangelizes.

Yes, with sudden death, even if someone happens to be Catholic, yet an obstinate sinner, such as divorced and "remarried" (living in sin), there's always that sorrow.  But we don't respond to that by claiming that Bergoglio was right and that one can be living in adultery and still be in a state of grace.

Now, as a Lutheran, your grandpa was likely validly baptized, so that does allow for the possibility (albeit very remote, naturally speaking) that he received some interior illumination and grace before he could no longer do so.  Time is of no consequence to God.  He can accomplish anything in a split millisecond of time, or can suspend time as needed.  Various saints (invoking God's power) have raised people back to life so they could be baptized.

And even in the case of a Jєω, as in 2Vermont's father, God CAN easily provide both conversion and the Sacrament, sending an angel to administer it.  St. Cyprian, who believed in "Baptism of Blood", stated that those martyrs receive THE SACRAMENT of Baptism.  I know that the Dimond Brothers call this out as a error.  But I'm pretty sure he meant exactly that.  He said that the blood supplied for the water while the angels pronounced the words (of the form).  So for him BoB was still the Sacrament of Baptism, except that blood was used instead of water.  Of course, Trent dogmatically taught that natural water must be used for the Sacrament.  But it's still interesting about what some Fathers REALLY meant by "Baptism of Blood," where for some of them it was not an exception to the necessity of the Sacrament but an alternate mode of administering it (with matter and form).

There's absolutely NO NEED TO POSIT Baptism of Desire even for emotional reasons.  Only those who lack faith feel compelled to do so, with one of their arguments being that God could be prevented by "impossibility" from bringing the Sacrament to His elect.  St. Thomas said of the pagan living in the jungle that, if they're properly disposed, God can (and will) send an angel if necessary to convert them.  What's to stop Him?  Impossibility?  If Our Lord God taught us that one must be born again of water and the Holy Spirit, there's nothing that could possibly get in the way of making that possible for His elect.  Even XavierSem who had been very much anti-Feeneyite, in the end, began to hold the opinion that God would bring the Sacrament to all His elect ... even if it meant using extraordinary means.  What need is there to speculate about substitutes for Baptism when something would render its reception "impossible"?  That's almost heretical, claiming that God is constrained by impossibility.  There's this general slur against Feneeyites that we believe that God is limited by the Sacraments, where it's really they who presume to limit God ... with "impossibility".  We simply believe that Our Lord will keep true to His word.  If He stated that the Sacrament is necessary for salvation, then you can be sure that He can and will get it to His elect.

I forget which saint it was, but there was a devout (apparently Catholic) woman who regularly received the Sacraments and then died.  It was revealed to the saint that the woman had not been validly baptized.  So he raised her back to life in order to baptize her.  While back, she stated that she had been without her "wedding garment".  That harkens of course to the parable of Our Lord in which He explained that those who show up to the banquet without their wedding garment will be cast out.