(and third I need to be careful wear I put my fingers on this laptop
In general there is not right or wrong answer to this. In specific there can be better or worse answers.
I deliberately allow my horses to eat during rides of more than hour. They learn the command "graze". I teach it to them because on a long distance ride they'll only have time to any appreciable grazing before we start and after we stop for the day. So every hour or more I will stop and allow grazing for about 10 minutes. I prefer to do it like this for the sake of keeping something coming into their stomach throughout the day. It's not a good substitute for 7 or 8 hours of grazing (a nice days ride on long distance riding), but it beats not getting to eat anything and I like it better than my other alternative of letting them stop and graze for a couple of hours after 3 or 4 hours of riding. I do during shorter rides of only a few hours more as a mean of training and conditioning them to eating like this. If I give one of my girls a grazing stop and they don't go to eating within seconds then I'll start them back down the road. The whole point was to train them to eat when told to "graze". If they don't take advantage of it then they can keep riding until I give them another chance (and of course I praise them when the do graze on command). Doesn't usually take too many rides before they figure out that they better graze when they are given the command to (it could be another hour before I give then the next chance).
That being said. If all I did was trail riding for a few hours a week I'd just allow them to eat when I wasn't riding. They'll have plenty of time to eat before and after the ride.
It comes down to the kind of riding you do and what your horse needs to be conditioned for. I can't see anyone making a blanket statement on it being best to allow your horse to graze while riding. I would completely disagree with the horse being allowed to decide when it's allowed to graze. If I'm out riding with friends for a few hours my girls can figure on being allowed to graze only if we all stop for awhile and take a break from riding (as if that's going to happen much
That PP makes a blanket statement about allowing horses to graze while riding doesn't surprise me. This is from someone who looks at training a horse as if all horses train the same way and what works with one should work with all (and if it doesn't then it must be the fault of the horse).