I agree, horses can sense/see things that we may not. I was not paying attention one time in town--trying to keep an eye on traffic, and my horse refused to cross a drainage grate. Lucky for both of us that he was alert. I still shudder to this day thinking of what might have happened, and I give him the benefit of the doubt when he shies at something.
Yes, do train Little Buddy NOT to browse while working or even on a lead unless you give permission. You will save both of you a lot of trouble later on. Going out on a lead with the express purpose of offering him grazing time is one thing. Snatching mouthfuls while walking on a lead should be unacceptable. It is very difficult to UNtrain that behavior. Lesson learned the hard way! By the time I discovered this driving forum in 2005, it was too late...