There is one significant difference between the Russell/Wilt/Bird/Magic/Isiah/Jordan eras and the era of today:
USA Basketball.
Prior to 1992, every four years a select few college athletes would play a few games with their rivals, then they would go to the NBA and never play with them again (save for a meaningless All-Star exhibition every year... and even those had
drama). Players primarily looked at their rivals as enemies to be defeated (a not-unreasonable stance).
But then, you had the Dream Team. Every 4 years, the best NBA players would get together and work towards a common goal over multiple games: winning the gold for America. And then you add the World Championships, and you've got even more games where players aren't seeing each other as hated rivals, but as important teammates needed to accomplish a bigger goal.
Look at the way the Dream Teamers gush over their time in '92. These were seasoned veterans... Hall of Famers, even... and they were overcome with a sense of comradery and teamwork. Throw a bunch of young twentysomethings in a similar environment and you don't think they'll become buddies?
Note that football has nothing remotely similar to USA Basketball and while baseball and hockey have analogues, international players are widespread enough (and of a high enough playing level) to diffuse most of the effects of the absolute elite players teaming up every few years.
If you want to get back to the old days and eliminate the occurrences of players teaming up for titles, one of two things needs to happen:
1) USA Basketball needs to go away
2) International competition needs to advance to the point where a few good American players teaming up is not a big deal