Ame did state that she had no intentions of fixing/re-starting the league. I don't know whether or not she minds US trying to re-do the league thing, so...
I don't know if any of the old leaders are still on the forums, but since I have never seen one, I doubt that's the case. That should be no problem, though, it won't be hard to find forum people that would be willing to take those roles.
Whatever system Ame used for the badges was for Pokemon Online only, so it probably won't work for Showdown. What we could do, however, is keeping track of it manually with the following:
-Players make their Trainer Cards as they used to with the old League.
-Players show their Trainer Card images to the leader they're battling.
-If the challenger wins, the Leader puts their Badge graphic in the Trainer Card image using a simple image editing program, then returns the new version to the player.
-The player puts the Trainer Card, with new Badge, in their signature.
This should work perfectly, but it is very vulnerable to abuse; What would block a player from simply editing in all the badges themselves? To solve this, we could use several methods:
-Add some sort of "watermark" to the Trainer Cards that cannot be removed by anyone except the Gym Leaders, enabling only them to put in the Badges (If anyone else were to edit the watermark, it would be visible)
-The leaders keep a log of the battles (time, challenger, win/lose) and compare the data to see if players actually have the required badges. Might take some time from every battle to check the logs and everything, but should work just fine.
-The Trainer Card image would be a special file type that only the Gym Leaders have the tools for to edit. Only Trainer Cards with that file type would be accepted to show before battle.
There are enough ways to "revive" the league. It might take some more time to keep track of everything due to the lack of Ame's system, but together, this could be very possible. With Ame's permission, of course.
Let's hope this becomes a thing