First off, getting any of the horses out of the arena & out of the way of the runaway(s) is not that easy. You cannot take the risk of having that runaway horse getting out of the arena to run amok through the barns or parking lot--and you don't have very much time to get the gate open, horses out & the gate closed before that horse comes flying around that end of the arena again. Evacuating the ring isn't a realistic plan.
I agree with Jill about how there are so many Mini owners who do not realize the possible dangers of driving their horses. It's easy to assume that because Minis are small they are easy to handle and a wreck like this wouldn't happen to them. Truth is, when a Mini gets spooked and decides to run like that, he can cause a major wreck too.
Tip--do not ever try to stop a runaway horse by getting in front of it. Stay well away if the horse is still attached to the cart, because a shaft tip can do major damage if a person gets hit by one. If the horse is free of the cart it may be a little easier. With a big horse if you do manage to get hold of a rein there's a good chance you won't be able to hold on--not if the horse is running full out when you grab hold. With a Mini--there is a chance. I've done it--not with a harness horse but one I was longlining--he spooked & took off & I had to let him go. He was racing through the neighbor's yard, then around the end of the trees, back along the tree line, down the boulevard & headed for the yard again. I waited for him to come past on his next pass, reached out & got hold of a rein, fairly close to the bit (no use grabbing it further back, behind the surcingle--you have to get it near the bit, ahead of the surcingle, so that you can pull him around)--and managed to hang onto him & stop him. He was just a little 33" guy, but it still took a good bit of strength to hang onto him & stop him.
In the ring, with other horses and carts, be careful about trying to catch the horse that is running--if you miss you may deflect him into the other horses--it may be better to just try and keep him on the rail, and let him run down a bit. Even that may not work--if he's determined to come off the rail & head for the other horses, you may not be able to change his mind.
I know that it wasn't the problem in this instance, but it concerns me that a number of Mini drivers will take a very green horse into the show ring. It truly isn't a good thing to decide just days before a show to get a horse hitched up for the first time, drive him a couple of times & then take him into the ring. That really is an accident waiting to happen, and it's a fairly common thing in Minis--again, I think because they are small and people tend to think that "small" means "safe"--you can manhandle them into doing what you want, so you can get by without a lot of training. A lot of times you get away with it, but some day somewhere someone won't get away with it.