I agree that this should be under a different banner than the old League, but I wouldn't call it impossible. As I've said earlier, I would love to volunteer as a leader, and also to help with organization if this gets rolling. It will take some manual work, yes, but it's not as bad as people make it sound. Here are some regulations that I think would make things more manageable.
1. There must be an order in which gym leaders can be challenged, based on the overall skill of the leader compared with the typing. (Some types are obviously just easier to beat than others.) We could do tiers of leaders, such as two or three gym leaders that can be challenged from the start, and then two or three more who can be challenged with one badge, etc. This would prevent leaders from being completely overwhelmed with challengers, which leads to my next point.
2. Set some kind of limit as to the number of challenges a leader will take per night. It may take some experimentation to find the right number, but I think this would work. This again prevents leaders from being overwhelmed, and it allows them time to finish up whatever concerns the real world may bring them as well. There would also have to be a time zone in which a leader is open for challenges. I imagine these would be specifically tailored to the leaders' schedules, so as to make it easiest for them to accept challengers.
3. For a badge system, leaders would be required to keep track of who beat them. This is by no means a difficult task. Simply write down the names of those who win in a notepad, or a Word doc or anything you won't lose, really. We could report the results to a main thread on the forums so everyone can see who has made it where in the league. With a limited number of challengers per night, this would be an easy task.
4. As for trainer cards, we could attempt to revive them with Ame's approval, but I don't think they're really necessary if things are being kept track of in a main thread. They would be a cool bonus, rather than a necessity. That could be looked into further.
5. And as I've mentioned so many times already, just keep track of things in one main thread. It would have a list of challengers and their place in the league, so it would be public knowledge as to who has beaten what leader. This removes the necessity of dealing with an automated trainer card.
6. We would need volunteers for leaders. They would need to realize that this is a strict commitment, and know that if they are consistently unable to deliver what is expected of them, they may be replaced by a leader who can keep up with the schedule. They would also need to show proficient aptitude with the type they volunteer for, or at least moreso than the other volunteers for that type.
7. League would reset at the end of every "season," which would be a predetermined length of time. I don't remember exactly how long the Reborn League seasons lasted, but we could use a similiar system. Those who conquered the entire league would be remembered in a separate hall of fame.
That's what came to mind initially for me. Like I said, it's a commitment, but by no means impossible, nor as hard as some make it sound.