You say that mockingly...but really, do you?
KD and Russ have made it to the Finals 1 time, and got handled easily by Lebron's team
Harden and Howard haven't made it yet as a duo (both lost in their other appearances)
CP3 and Blake haven't made it yet
Wade and Bosh aren't gona make it back without Lebron
Melo never made it
Bottom line is people hate on Lebron, but there's Hall of Famers who never even sniff the Finals, and he's been there 6 times, the clear cut leader everytime. No one has ever had the expectations, and therefore the pressure, put on them from the time they were 17 and still managed to even come close to living up to them. It's a big difference when you have the freedom to fail and not be criticized for it. No one else can relate to Lebron's career in that sense.
Jordan won 6 out of 6 that's incredible (he also had Pippen in his prime and didn't win 1 until he did, Lebron had 1 maybe 2 seasons with Wade on the tail end of his prime and Pippen might have been the better player all around and was certainly the better defender, 2 long wing players that could defend 1-5 is a crazy combination). Jordan's accomplishments don't cancel out what Lebron's done and don't mean that what he's done should be mocked. And Jordan, just like Kobe, Duncan, Shaq, Bird, Magic, etc. all had the freedom to fail and play loose, miss buzzer beaters and be commended just for having the balls to take them, Lebron never had that luxury.
As for his Finals appearances, he was only 22 in his first trip against the Spurs and the Cavs had no business even being on the same court as them, and wouldn't have been there without one of the all time great performances by Lebron in the ECF against the Pistons (game 5 will be remembered as one of the greatest playoff performances in NBA history). He underachieved against the Spurs, but they couldn't of won even if he put up 45 a game (still averaged 22 7 & 7). Against Dallas he choked. He handled his business the next 2 times, giving Duncan his only Finals loss, and KD, Westbrook, and Harden a loss in their only appearance. And against GS he had another all time great performance and just didn't have enough help all around, with a healthy Kyrie he's most likely 3-3 right now.
Curry's incredible and might get a couple more rings, but then again he might not, and it'll most likely take more than his body can handle to get to the chip 6 times in his career, he's already 27 with a history of injuries and a relatively frail build as professional athletes go.
Every team Lebron's a part of becomes an instant contender. The Warriors were built perfectly with Curry being an amazing piece of a larger whole, that showed they can win whether Curry has a great game or not, again a luxury Lebron's never had. No one, including Curry, can walk on any team in the league and take them on a deep playoff run except Lebron.
As for the premise of the thread, it's completely wrong. Curry can take the mantle and has already built the foundation to do that, and if he doesn't Anthony Davis has the potential, Westbrook too, there's a handful of players that have the ability to be the guy the NBA leans on. They don't have to be as unique and dominant as Lebron to take the mantle as the face of the league.