Hi Chris,
First, I understand how hard it is to come up with an ideal run order! Even if you assign competitors to separate rings, you still have to worry about when your equipment (weaves and tunnels especially, it seems) is going to be needed, and it is not easy.
That being said, as a competitor, I have a strong preference for 2-ring shows that are run as a 2-ring, 2 trial format where you stay in the same ring all weekend. It makes things SO much less stressful, not only for exhibitors, but for workers, gate stewards, and ring stewards. I've also found that I, and a lot of the folks I trial with, are much more willing to volunteer in the ring when we know that the possibility of conflicts is gone.
Whenever I've been to shows that run non-jumping classes as "come when you can, don't worry about level" classes, it has always, without exception, turned into chaos. Some folks try to keep up the posted running order, but that only works for so long. Others get worried about conflicts, and try to jump in line, or argue with the gate steward. I've experienced lines of dogs at least 20 dogs long waiting for tunnelers... it's madness. (Anymore, if I know or suspect that a class will be run that way, I get my husband to take out my dog before we start walking the course, get her warmed up, and then get in line so that I can be right at the start of the line, before the insanity sets in).