The arguments for background checks generally have to be split into two separate classes of people.
The first is the mentally ill. Intuitively it seems desirable to say that someone undergoing treatment for e.g. depression shouldn't buy a gun. The problem here is the massive perverse incentive. If you're pretty depressed but you're not inclined to forfeit your ability to buy firearms, you now have a significant incentive to avoid seeking treatment. At which point you can still buy a gun but now your mental illness is going untreated, which is very worse than where we started.
The second is career criminals, i.e. people who have already been convicted of a crime and want to commit another one. The problem here is that career criminals... don't follow laws. If they want a gun they steal one or recruit someone without a criminal record into their gang etc., both of which are actually worse than just letting them buy one.
On top of that, when people get caught, prosecutors generally try to get them to testify against other criminals in exchange for a deal, who are then going to be pretty mad at them. Which gives them a much higher than average legitimate need to exercise their right to self-defense once they get back out. And then you get three independent bad outcomes: If they can't defend themselves they get killed for snitching, if they acquire a gun anyway so they don't then they could go back to prison even if they were otherwise trying to reform themselves, and if they think about this ahead of time or are advised of it by their lawyers then they'll be less likely to cooperate with prosecutors because the other two scenarios that are both bad for them only happen if they snitch.
Meanwhile the proposal was only ever expected to address a minority of the problem to begin with because plenty of the people who do bad things can pass the background check. And if you have a policy that doesn't even solve most of the original problem while creating several new ones, maybe it's just a bad idea?