Make no mistake. Winning 6 championships is a very impressive feat. But doing it for 1 organization, rather than multiple organizations, could be a reflection of finding yourself with the right mix of players and right coach. It doesn't require any adaptability or achieving chemistry with a different core. It doesn't require a player to win under different cultures and management structures.
When you win championships with 3 different organizations, that meant you were able to lead 3 different rosters and different cores to a title. That means you were able to win regardless of the coach and supporting cast.
There's some truth to that. However, one also needs to consider that before free agency opened up it was a lot harder for star players to move around. Which meant that if a guy had a winning situation, it wasn't a given that the superstar switching to another team would 1) be very easy to pull off, or 2) allow for a team to be built around him relatively quickly. Lebron couldn't have team hopped like he did back in the '80s for example, it just wasn't something the cap rules made easy. Most big name movement back then was only via trade. In all likelihood, Lebron would have been stuck in Cleveland until he forced a trade via holdout or something.
Before the 90's, how often did a top 5-10 player even become a free agent? Not often.
When Jerry West started trading for specific contracts and accumulating young assets in the early 90s in anticipation of cap dumping and making a run at Shaq 2-3 years later, that was a big shift in how NBA teams were built. Before then, you never saw teams do anything of the sort. And why would they? Guys like Magic, Bird, MJ ... they never hit the open market anyway under the old rules. "Oh wow we have all this cap space, guess we can sign ... uh ... Tom Chambers."
If the current rules had been in place, then you might have seen some other great players go to other places and build up new winning situations. Perhaps Magic takes a gig on the big stage of NY with Ewing. Maybe Larry goes home to play in Indiana. Perhaps MJ gets away from the two Jerrys before he ever won a single championship with the Bulls (Scottie almost certainly would have left). But those weren't avenues open to them back then.