The NRA's clout: It comes from the people
Gun rights are not popular in America because the NRA is so politically influential. The NRA is politically influential because gun rights are popular.
There was no NRA when James Madison penned the Second Amendment, and the belief that the Constitution protected the individual right to bear arms pre-dates the NRA. (UCLA constitutional law professor Adam Winkler has detailed how the 14th Amendment was proposed and ratified in part to guarantee that former slaves could not be disarmed, as the Ku Klux Klan and many Southern state legislators were trying to do.)
Many on the left believe that if the American people do not agree with them, then the people must have been brainwashed by powerful forces. They cannot fathom why their opposition to guns and gun rights is not the dominant view. It must be that evil NRA.
But the NRA did not write the Second Amendment or make it popular. The group would have little clout if it did not have 4 million members and the support of millions of non-member gun owners who see a need to become politically active to protect their right to bear arms from liberals who openly advocate eroding it.