There are some options, just the biggest obstacle is that there is such a short turn around between getting your qualifiers and then getting tickets for family and teams then for the general public.
Some ideas I have had or heard from others.
1. Assigned sections instead of seats. I assume this is possible with enough "service" fees on Ticketmaster, but it would be great for teams and fans to converge on a specific section instead of exact seats. I really miss the pockets of fans from each team that cheer loudly together.
2. Teams get X amount of tickets to sell per qualifier beyond the parent/school tickets. If it was say 10 per qualifier they would get the 4(well should be 4) for family and 4 for school, plus 10. This would mean just over 4000 tickets which is a lot. The unused ones would be given back. The hardest part would be the turnaround time to get them to schools and get them back into the pool if not used.
3. Only all session tickets.
4. Open All-Session tickets on Wednesday and single session tickets on Thursday.
The assigned seating isn't going away, general admission is bad and makes for a tremendously less enjoyable experience.