Ignoring for the moment of it would work on the real world or only on paper, the premise is this: The more guns the easier it is to get those illegal guns.
All guns were legal some at some point. It isn't like these people are hand crafting these things in basements and abandoned buildings. At least of any are they are in numbers so miniscule that they are a blip in the statistics.
These legal guns become illegal in many ways:
I'm sure there are more, but that should cover most situations.
Because I'm trying to explain this neutrally in an on paper method to explain the viewpoint, and not meaning to advocate, I'm going to switch to using hammers as a stand in for guns in a likely failed attempt to take a step back emotionally.
Lets say you were trying to get sledgehammers off the street. The illegal hammers can't be made more illegal. The illegal ones will break. They will get confiscated during arrests. If you can keep making them rare enough they will gradually get priced out of increasing number of criminal's budgets.
To do that you need to limit the creation of new illegal guns. Background checks are one way, but won't prevent thefts or two neighbors selling to each other for cash since they don't see the point of going to a dealer as a intermediary. Even well meaning background checks fail. So limiting or banning sledgehammers is the most absolute way except maybe a biometric using "smart" sledgehammer.
With the end of legal sledgehammers the legal pool of them will start to dry up first, but so will the illegal pool. It'll be a process that takes time.
Of course there are problems with this, but that's the gist of the mindset.