I agree with Doe. As for setting the rule being demoralizing, well, one has to make a change sometime. If the horse becomes resentful about it , don't worry or try to hug him and say you're sorry. That only makes it more confusing. Let him have his opinion about it but keep to the rule. He will adapt.
My rule is if the horse is on a leadline that is in my hand, he is not allowed to graze . . .EVER! Even if I am standing and talking to someone, not leading him from the pasture. He is not allowed to eat when I am in the saddle, either.
BUT, if he is on a leadline and we are in a nice green area and I have the time, I will drop the leadline to the ground and let him eat. If he steps on it, he will only learn how to release himself from the pressure which is a good lesson for the hrose to learn.
Once I pick up the line, if he goes to eat, I kind of throw the line at his head (send a ripple down the line which makes the knot at the end bump into his jaw). I much prefer a rope halter and anice heavy rope leadline to the more traditional lead and webhalter. One has much better control.