Real answer:
Russell, because he is the league's greatest winner. Also, because he was so innovative. First black coach. First black leader of a team. And a major advocate for civil rights. And he was also one of the greatest rebounders and defenders ever.
Bird and Magic because besides the talent and accomplishes, they led the NBA out of the disaster that was 1970's basketball with there being two leagues and rampant cocaine and other drug problems affecting the game. They saved basketball. Ratings soared, as did revenue and basketball overtook hockey and baseball as the country's 2nd most popular sport. All because of Larry and Magic. They were that influential.
Then, of course, Michael who took everything from Larry and Magic and built on it. He also became the best player ever. The greatest shoe salesman ever(after Bird and Magic started the basketball shoes revolution and branding). And a culture changer.
Those are your NBA Mount Rushmore players, because reaching those heights means more than just counting stats.
That's a great list based on your criteria, but I disagree about your swipe at statistic based evaluations being purely counting stat based - the use of impact metrics that actually attempt to tease out how a player affects the scoreboard have been gaining popularity the past decade. These statistics are hardly just counting stats, they're literally trying to quantify what Russell said in his statement about the "other 45 minutes of the game". Obviously there are massive outliers in such metrics (as with any other statistic), but you can form a very convincing list when you use those stats as a tool to aid what you see on the basketball court.
When determining the greatest that is on the Mount Rushmore, IMO, you want guys that did more than just ball. You want guys that changed the league for the better and did so in ways off the court as well.
It's fine to say LeBron is a better basketball player than Bird or Magic, but to be on Mount Rushmore of the NBA??? Bird and Magic completely changed the way basketball was played and perceived in this country. Ratings took off and better TV deals happened because of Bird and Magic. Shoe branding with Converse started with these guys. They saved the game. So because of that, I don't care that some advanced stats might say there are slightly better players than Bird and Magic, they weren't more important to the game.
They changed rules to account for Wilt. MJ became the first global icon of the sport. Lebron has taken MJ's iconness to new levels especially his use of social media, his media empire, etc. This idea that the other great players didn't effectuate change is just strange to me. So elevating inferior players for those reasons also seems strange to me.
A bold claim to assert that LeBron has ever been a bigger icon than Jordan. Not during his career thus far, 21 years after Jordan's retirement (
) Potentially he's a better basketball player. Not an opinion I share, but a valid opinion.
Anyway:
Russell, Wilt, Bird & Magic, MJ.
Kareem, LeBron, Dr. J, Oscar.
Iverson, Shaq, Duncan, Walton.
Not the best players, but the players I'd most like to see have their faces carved on a big mountain somewhere.