Unquesionably, MANY people who are prevent from legally owning a gun or purchasing a gun would and could end up going the nefarious route to purchase one; but I think the notion that just anyone would, as if there would be 0 difference between the number of people wrongfully obtaining a firearm if there wasn't background checks and if there are, is a flawed one.
Are we really concerned about "just anyone", though?
I agree that we would see some difference in the number of people illegally purchasing guns if the "loophole" were closed, but are those really the people we're truly concerned about?
I would think that the reason we'd want to keep guns out of unauthorized hands would be to prevent the many mental-illness-related mass shootings that seem to have become the fashion, and to prevent other firearm shootings and murders, and armed robberies, and firearm-related assaults, and drive-by shootings, and things like that.
And I wonder if the guy who is willing to carry an illegal gun around with him on a somewhat regular basis, and use it to rob, assault, shoot, and kill other people is the kind of guy who would be deterred by an administrative loophole closing when, as we know and as you've suggested, there will still be plenty of illegal guns available for illegal purchase, they'll just be a little bit more illegal and a little bit harder to obtain.
To use your drug example, the guy deterred from purchasing a dime bag of marijuana while on vacation because of the social consequences of getting caught with marijuana in his system isn't the kind of guy who would buy an illegal gun and then use it to go out and commit strong-arm robbery or home invasion.
He might buy an illegal gun if he thought that nobody would be any the wiser but it's unlikely that he'd use it as a tool for mischief, or worse.
If we had to think of a drug use analog for the kind of person who would buy an illegal gun and then use that gun to do still more illegal stuff with it we're looking more at the 10 bag a day heroine addict, and we know that existing draconian drug laws are little deterrent for that kind of person.
To me this whole argument comes reminds me of the old adage that, "locks were made to keep honest people out".
If you're willing to respect a lock, or social pressure to not use drugs, or the closing of an administrative loophole, then you're not really someone to worry about.
But if you'd kick down the door to rob a place, or if you're the kind of guy who smokes crack despite the numerous draconian laws prohibiting the drug, then you're also in the class of people who would use an illegal gun to commit even greater crimes, and if locks and drug laws aren't preventing such people, at all, from doing their thing then what realistic expectation can we have that gun laws would make a difference.