I don't mean to defend the NRA here -- as you'll see from my other comments while I recognize and appreciate the stated role they perform, I'm not their biggest fan. Regardless, that particular allegation doesn't carry much water for me.
Speaking in broad strokes, every time there is a shooting, it seems incumbent on every gun owner to bear the responsibility for the bad action. Because we support the right to keep and bear arms, we are often cast by the media as the very reason that whomever was able to effect whatever recent spate of gun violence occurred, and must all be held accountable. Whenever politicians hold session, they urge to DO SOMETHING, and rarely care what that something is so long as they can show that they have been responsive in passing legislation that may positively affect this situation in a future outcome, even when it's unlikely to have any effect.
If you're incapable or unwilling to cede to things like increased background checks, or unable to sign on to a blanket ban of muzzle shrouds (as our politicians so often request), there aren't a lot of politically viable options for "doing something", which casts the second amendment lot as obstructionist. What's left? Well, increased sentencing for gun felons. These increased sentences are regrettably more constitutional, because they at least are the result of due process, and do not infringe the rights of law abiding citizens.
Are they good policy? I would argue not, but if you're faced with the nefariously binary options of removing rights from the law abiding vs. removing rights from proven criminals, it is easy to see how they routinely choose the latter.