I say yes. Any animal who loses a limb or body part goes "off" in structure. As an owner and handler of high level agility dogs, I understand the importance of canine structure. In agility, if a dog is sore in one leg, the leg on the opposite side will also show soreness as the dog uses it more to help keep weight off of the sore limb. So, if the dogs left front leg is injured, the right back leg will also be sore. If the left back leg is injured, the right front leg will be sore.
An average house dog owner would never notice this affect, and in fact, probably wouldn't notice the original injured leg. But because of the intense rigors of the sport, agility owners are constantly watching their dog's gait for injury that will need immediate attention. Canine chiropractors and canine massage therapists are in huge demand at agility trials as competitors work to keep their dogs' structures in-line and limber.
We also make sure our canine athletes are well exercised using various canine exercise products. A very popular product is the "peanut" exercise ball. I've written a review hub of that product. These products help our dogs develop strong muscles to help prevent injury in the first place. Most agility dogs competing at the higher levels are on some sort of regular exercise program. My agility dogs are on a regular exercise program. They also see a chiropractor when needed, and proper warm up, cool down and massage is administered before and after each agility run or practice. My dogs also see a canine sports vet who specializes specifically in structure ailments in the dog.
I would recommend you examine the importance of canine massage and perhaps chiropractics for your dog. Also, start Googling and looking around at the importance of strength exercises for the tripod or injured dog. There is a ton of excellent information out there, but you need to go searching for it.
Yes, other than keeping your dog's structure in good shape, there will only be a few nods to his handicap. Dogs are amazingly adept creatures. However not to acknowledge that your dog has a structure deficit and not doing anything about it could lead to early arthritis and other pain later in life. It does in agility dogs, especially if they incur any important injuries during their careers.