Don't blame the dog.
I have been hiking the trails around Flagstaff for over 45 years with my trained dogs running at large. These dogs have a greater respect for the trail system than runners, mountain bikers and equestrians. They don't tear up the trails with their paws. They don't charge up behind you expecting you to get out of the way and they don't leave manure on the trail. If they leave feces on or near the trail, I bury it or pick it up for later disposal. Most of all they do not go after any of these other trail users.
An ordinance that prohibits dogs off lease is unfair and unenforceable. Is the County Board of Supervisors going to fund personnel to be on every trail to ticket people with dogs running at large? Will the Forest Service post rangers at trailheads to enforce the leash law? Are dog owners expected to keep their dogs on a leash or in a suitable enclosure while camping in the national forest within Coconino County?
If a dog is under owner control, the dog should have the ability to run at large, especially in the forest on or off a trail. If a person has a vicious dog off leash, then it is appropriate and wise to notify the responsible government official of the owner with a vicious dog. It is also appropriate for an individual to request the owner put the dog on a leash.
Some of the most vicious dogs I have come across are those in training as hunting dogs. These dogs are allowed to be off leash while training according to the ordinance. Dogs participating in field trials and obedience training are also allowed to be off lease. I guess my dogs will always be participating in field trials and obedience training if anyone asks why they aren't on a leash. Well trained and dogs under owner control should be able to run free. Don't blame the dog, blame the owner.