You are making an incorrect assumption.
Nothing stops what you call "Undesirables" from obtaining firearms now does it? All Gun control does is restrict the law abiding. Prior to 1968 the Second amendment stood on its own. Firearms were sold at hardware stores and through the mail.
The other problem is who gets to define who is undesirable? Andrew Cuomo did it in 2013. Meanwhile, the rule of law says something about "shall not be infringed."
I was born and mostly raised in a 3rd world country. They are on the list of the top 10 places to die from GSWs. It is very difficult to get a gun there, yet the criminals are awash in guns.
NEWSFLASH: Sometimes our politicians are bigger criminals than the populace. I WiSH I could give examples, but alas I don't have any.
We are being sucked in, into thinking that background checks reduce crimes. It really does not. Insane people (long term or heat of the moment) are going to still exist and still be able to fool any system we put in place. Likewise for those who APPEAR to be upstanding - that's usually the person buying whores and screwing goats (I may have that switched around - apologies! )