That's not quite the story. Or it is, but only in the broadest of strokes.
The real story boils down to Harlan Carter, and his bullshit.
See, prior to 1968, the NRA did get involved in politics. They were consulted on gun regulation, and even had a hand in helping to write the NFA and Gun Control Act of '68.
This pissed off Harlan Carter, who was a racist, unrepentant murderer, and former head of the US Border Patrol, into staging a coup for the control of the NRA leadership.
He then instantly started running the NRA as the propaganda arm of the gun manufacturers, and was paid handsomely to do so. It's also when the NRA "got into politics".
It's also when they somehow collectively forgot about the second half of the Second Amendment.