I think Picard was willing to sacrifice himself to save the kids. He's an officer who signed up for a risky job - they are not, and also they're kids. I think he thought that going with them would slow things down enough to add unacceptable risk for the kids. And they did end up spending a bunch of time cobbling together an apparatus to move Picard during which the lift could have fallen.
When the kids refused to go maybe that changed Picard's calculation: the advantage of going without him diminishes if they use up time arguing. Or maybe it's TV writing.
But maybe Picard wasn't certain that the lift would fall. Or maybe if he'd stayed he would have managed to pull out a Picard move to save himself at the last second - you know, the kind that's easier to do when there aren't kids watching. Or maybe, as far as he knew someone might rescue him in time. But yeah, he probably would have died, and the kids' mutiny was the only out that let him save himself while also trying to be noble.