I've noticed that often there's an active ongoing tournament with dozens of players, but there's no link to it from the main page. e.g. currently there's a superblitz arena with 37 players, but the "Open tournaments" section only displays tournaments that haven't started yet. Is this a bug?

The most reliable way for me to find which tournament people are currently playing in, seems to be to click on the game currently on "Lidraughts TV" and go to the tournament from there. But this is obviously not ideal.

On the homepage it shows upcoming tournaments and active tournaments that are less than 2/3 done, so it depends on how long a tournament has been running.

I think the best way to find active tournaments is from lidraughts.org/tournament, there you see all running tournaments and how many players are in it

You could consider extending this based on some heuristic — 1/3 of a tournament can still be pretty long, based on the tournament. If there's a big tournament with 100 players going on then it's obviously more relevant than an upcoming hourly arena, even if it's near its end already.