Now, people are starting to question the position of the NRA that *anybody* regardless of who they are should be able to procure guns
Not their actual position, although waving blanket, false statements like that around is what passes for political discourse these days. Actual fact: the current background check system was actually strongly supported by the gun lobby: people who are convicted felons or legally declared mentally incompetent don't have second amendment rights. Or many other constitutional rights, say for example, voting. The current argument (causing the House to behave like the dysfunctional third world legislative clique it apparently actually is) is over the "sounds good!" legislation of "people on the terror watch list shouldn't be allowed to buy firearms". Hmm. So, a law in which denies something listed on the bill of rights to people on a secret government list, who can get on that list simply by someone voicing suspicion, with no procedure for getting off the list (or even knowing if/why they're on it)? Pick anything else that's a legal right (voting? free speech? Self-incrimination? Illegal search and seizure?) and swap that in for "gun ownership" in this scenario and watch everyone across the political spectrum freak out. We tried something like this in the 50's with McCarthy when the enemies were Commies instead of Radicals, and are universally ashamed of that fact in hindsight. Of course the NRA should be objecting to this. I'm shocked that the ACLU, for example, isn't too.