Other.
There is one simple solution to this fiasco.
You NEED a central booking system. From there priority can (and should) be given to people entering all meetings first, enough meetings to complete the series second, and to anyone else third. Booking in needs to open well in advance and close a couple of weeks before the first round with confirmation e-mails sent no later than 1 week after closing.
After that it doesn't matter what format is run (as long as the format was clearly advertised before booking in opened). People will have been able to make an informed decision from the info provided, with no "first come first served".
Can I close the thread now?