The whole purpose for a check is to artificially lift the horse's head. This results in a hollow frame, instead of a truly comfortable well-working frame. I personally think that if people really knew the mechanics of what is happening muscularly and structurally to the horse, they wouldn't use it, but since it is required by some breed organizations, and is a part of some harnesses out of the box, they think they need it. You can lift the horse's head without the check, but it requires work and that the horse be conformed to do it!
As far as not snatching grass, that training starts away from the harness and cart. Minis should not be allowed to graze in hand unless permitted. You shouldn't have to walk your mini and wait for them to get their mouthful. When they go to throw their head down, you snatch it back up with the lead and sometimes your foot in the nose for very persistant horses. Then if they try it in the cart, you may have to snatch their mouth, usually one rein at a time works better, when they go to put their head down. I would equate it to the pull you would give a child that is holding your hand, but wants to dart into traffic. You are going to "snatch" them back.
Actually, when we didn't know any better a number of years ago, we almost had a bad accident that involved a side check and a grass snatcher. He threw his head down to snatch grass (we didn't let him, he was being sneeky), and when he came up, the check was under the shaft. That made the check tighter, and he almost reared up and over. The show manager happened to be near and was able to pull the check out quick. The check came off right then and there, and we haven't used it since. As far as the snatcher, you have to be quicker than he is. When he starts to put his head down to snatch, you pull up with a growling voice command. Pretty soon, you will just have to use the voice command, because they want to avoid the pull on their mouth. Be "mean" if you have to! It really is a safety issue, because the rein can get under the shaft, too, and then you have no control.
Myrna