Firstly, the members of the fellowship of the ring had to gather in order to destroy something evil that was there and existent. This automatically turns them into the good guys and hence entitles them to compassion and sympathy. On the other side we have a self proclaimed "Big Three" along with a bunch of mercenaries who did not need to gather for anyone else's but their own good. This automatically turned LeBron James, Dwyane Wade and Chris Bosh into the evil itself (everybody knows that we love the underdog and that's why it is the smallest and most insignificant guy from the fellowship that has the biggest impact in Tolkien's book - Samwise Gamgee - yes him, let's not forget that Frodo turned away and if it wasn't for Gollum he wasn't going to destroy IT... What is more Frodo is not the ultimate underdog - Sam beats him by far as he is a simple gardener). But for now let's keep Miami's big three as similar to the fellowship rather that straight forward associating it with Barad Dur. This will allow for some more exposure to the reasons behind Miami's struggles on the basketball court from a fairy tale perspective.
Secondly, the fellowship of the ring knew what difficulties it had ahead of itself (orcs, goblins, traitors, evil wizards, trolls, nazguls (disturbing but interestingly the grammar spelling suggestion to replace nazgul was "angel"...), an evil eye etc.). Unlike Miami's Big Three they were also aware of the existence of two towers that were in their way to the destruction of evil (do you sense where this analogy is going - purple and yellow army of Uruk Hai and some Green goblins...oops... that sounds a bit Spider Man but its not!!!). For Miami's Big Three it seemed all about getting together and hopefully everyone else would just allow them to get to the finals of the most competitive league in basketball today. Following this type of thinking Frodo and Sam could have just taken the shuttle bus to Mount Doom, or been given a lift by one of the Nazguls under the protection of the evil eye of Sauron (just to make sure no orc gives them any trouble) and thrown the One Ring into the fire.
Thirdly and finally, all of the members of the fellowship did not have a status of heroes before their quest while LeBron, Wade and Bosh were all quite used to being heroes for their clubs. They all simply took a step down by hiding behind the excuse that they are team players and can share the floor. Let's think Tolkien again - we all knew who was the alpha male in the fellowship - yes the elf was slick and smooth, the hobbits as mentioned were public favourites due to their underdog status but everyone expected Aragorn to sort everything out. All, three players of Miami's Big Three were used to being Aragorns for their teams and there is nothing wrong with that - that is who they are. A basketball team needs an Aragorn and I am telling you none of Wade/LeBron is ever going to be willing to be the Gimli of an Aragorn. Bosh is a different story - he was just a hero in relative terms in Toronto - a Gimli among the Shire folk.
But they took on a path that would eventually lead them to a ring - that is certain. But they will never ever get to be the good guys unless they figure out who will sit on the throne of Miami. Will the King (LeBron) return (can you feel the threequel) and occupy the role he had in Cleveland? Or is the original Stewart of Miami (Wade) keep his place on the throne. It is an "either-or" situation and I want to remind you what happened to Denathor when he heard the real King was coming.