Maybe start by checking why there are so many no-shows in the ASTs. Is the thing requiring players to join the lobby broken?
Then, a way (or ways) to help set up custom challenges between other people!
This seems easy, and from what I’ve seen it could be useful for lots of groups that organize offline but play the matches on OGS. If you include a startPaused option, that might be just enough for them to set up a game ahead of time and start playing on some schedule.
It would be great to support some radically different tournament styles, and I think there’s an easy way to get it: separate out the organization of the tournament from the rest of the site. Let anyone code their own style, starting matches via an API, with only a bare minimum on the back end to support a registration system and a results page. Then we could have arenas, ladders with live matches, etc, without being tied to a particular buggy implementation that nobody understands.