If you watch CG it is not about dog liking crate or running towards it with success. It is about criteria and self control.
Bob Bailey attributes training problems to: timing, criteria or rate of reinforcement. Not maintaining sit is a criteria problem IMHO. The way Crate games are build criteria is crystal clear- do not move until released. So the word stay is eliminated by itself as it is always sit maintain, down maintain till released. Chris Bach teaches that as well, you might want to rent her tapes either from Tawzerdogs or Bowwowflix.
Inability to maintain sit under stress is just a feedback from the dog that criteria of sit is not proofed enough or understood. You need to be creative and come up with enough stress outside of the ring so criteria is maintained no matter what. For example if I want to train/ proof dumbbell hold I will test understanding by asking dog to hold dowel and then put a piece of steak into his mouth. If he still hold the dowel I think it is a good sign he understands task. If he drops dowel in order to chew the steak, he does not have an understanding of the job.

I do not particularly care about competing in obedience but I like messing with dog brains

Today for example I felt lazy so for training they were into sit on the mat while I sprinkled handful of the kibble around them and around the room. They maintained sit which was about 90 seconds and then on release were allowed to run around looking for kibble. It was fun for everybody.
As Justicemom mentioned, I would eliminate physical problems with not maintaining sit in the first place of course.
This series are about agility but advise is very clear and useful for dogs in ANY sport.
http://www.runorstop.com/