In my opinion, treats do not hurt anything. Depending on the horse, I will give them carrots or apples or some other type of horse treat. Cow Cake works well with my horses. I do not ever give them sugar cubes, they just don't need the sugar. I have not had any problems with horses getting pushy for treats. I have taught them to take carrots out of my mouth, my pockets etc and never had a problem. Keep in mind that the horses have all been mares or geldings or young colts, yearlings or less. Would not do that with studs, but even giving them treats is fine if you know the horse and don't let them get nippy as that is a normal tendency for a stud.
Did the vet say why she didn't like your horse? From the picture I don't see a major problem and if you are happy with the horse and the horse works for you, then unless it is a health or lameness issue or the horse is not suitable for what you want to do with it, then I wouldn't pay any attention to what that Vet told you and I would change Vets to boot.
Got a horse, got a dog, got God, got enough.