I always allow grazing and have never had any issues with it (for the last 13 years or so). It is very helpful with barn-sour horses who do not enjoy trails if you let them eat at the farthest point before you turn around.
It seems to help nervous horses "relax" as they are not so worried about everything that is going on around them.
I do not ride consistently and I ride by myself a lot. I might trail ride once every 3 months or so, sometimes every 6 months, sometimes less than that. I want my horses to enjoy going out, and I do not want to worry about spooking or separation issues. The grazing always makes it a happy experience and my horses love to go out.
I've never had any issues with them pulling on me or refusing to put their heads up. In general, they will grab a bite and keep walking. My horse's are sensitive so I never have to get after them.
When teaching a weanling to leave the yard by themselves, I always hand walk and allow them to graze. I think it is important to get them out and about at a young age. With babies I do carry a whip in case they get in my space or get nippy. My trainer taught me that method, and always had great success with her babies. This way they get exposed to traffic, dogs, and everything else! By the time they are 2 yr olds, they are easy to trail ride as they have already had so much exposure.
I don't know why people view it as such a bad thing.