If he goes well in the snaffle, there’s no reason you couldn’t use that for every day riding. It just kind of depends on the horse. He may even work the barrels nice in one. There’s no “rule” that says you need something strong for running, but it just depends on the horse.
For my younger horse Shotgun, I primarily ride him in a snaffle to keep him legged up. He will be 7 this year. I do a little bit of everything with him, including showing. So I do also ride him in a short shank curb (double jointed mouth) when we are showing or doing reining. For barrels, I can run him in a snaffle too but lately I’ve been running him in a Jr. Cowhorse with the roller center. He seems a little “quicker” in his responses on the turn with it. But he does work nice in a snaffle too. He’s still learning - so far his personal best is top of the 3D … and he hasn’t figured it out yet LOL.
For my other horse Red, he hates snaffles. He’ll go in one if I ask him to, but he’s not a fan. So I do most of our regular riding in a simple short shank curb with a dogbone center. For barrels, I also run him in a Jr Cowhorse with the roller center. He’s usually 2D (3D if I make a mistake) but gets a nice 1D run once in a while. For training for reining, I’ll mix it up. Here and there I’ll ride him in a correction bit or something a little stronger, then come back down to our usual bit. I’m always “messing with him” so to speak, so I try to keep him light and on his toes.
For your horse, the head tossing could be a training thing (maybe why your trainer thought a million dollar bit would work for him … but that’s a lot of head gear). Or he maybe really didn’t like the Ed Wright bit. I’d try a snaffle on him, or whatever other bits you happen to have right now. Do a few days on each bit and just play around with it and see what he seems to like and/or go best in.
If he tosses his head with everything, then I’d be inclinded to think it’s a training issue and not so much the bit. (provided his teeth are fine, saddle fit are fine, etc etc)