I saw the groom very sharply elbowing her in the chest today for eating grass, she tells us we are too soft and pushes the horses around.
I call this "Truffle Hunting" when the horse decides to dive for the grass without permission. You're walking along and there's a choice looking bit of grass and the horse dives down to eat and plants his feet. Depending on what's going on it can be mildly irritating, downright disrespectful or dangerous (ever try removing horses during a fire?). Eating grass isn't a bad thing in and of itself, it's diving down and stopping without being told it's alright that's the problem. When you're walking along, just dawdling and letting the horse hand graze, then no, I don't think a jab to the chest is appropriate. If you're moving a horse from one nice grassy pasture where they've been grazing to another pasture where they're going to graze and trying to move along at a business like pace, then truffle hunting is disrespectful and a jab to the chest or a tap under the chin from the toe of your boot is a reminder that it's not ok to take that decision on themselves. The difference is, you have made the decision in the first instance and the horse has made the decision in the 2nd.
My order or reprimand goes something like this: I halter the horse and lead him out, headed from one pasture to another. The horse goes to dive and I give a quick pop of the lead line, I NEVER use a chain shank unless the horse is an absolute PIG, before they can get their head down to the grass. 9 times out of 10 that's sufficient with a horse who has already been trained to have manners. If the horse tries again, I pop harder, most likely a little sideways so the snap of the lead line 'bites' them on the chin and if they persist after those 2 much more polite requests to behave, then I tap them very firmly (no not kick) right under the jaw where the jaw bones come together under the chin with the side of my boot. Should there be a 4th time, then we go back to some very basic ground manners training because obviously I have missed something along the way.
A horse, and pony's are horses condensed, will test to see who's going to be in charge. If you fail to step up and be the leader, then the horse will step into the void. When that happens it's usually a pretty poor outcome because, as someone already mentioned, reasoning and long term planning are not a horse's best faculties. The human needs to be in charge and remain in control at all times, especially with ponies as they can be some ornery little things. A pony is not an overgrown German Shepard or Great Dane. For one thing, one is prey the other predator, and both think wildly differently from each other.