I would skip the joint supplements. Better bang for your buck going with something like Adequan, Pentosan or Legend. They work via different mechanisms, so which one to try first, or switching to a different one is a conversation to have with your vet.
As far as joint injections? I would never inject a joint without an x-ray, especially if the horse is sound. About 12 years ago one of my horses was mildly unsound on his left hind. Tried adequan with no improvement. After a few months we took him to a specialist for a workup, with the assumption that we would end up doing hock injections. X-rays were clean. Turned out it had been a hamstring injury and a round of shockwave had him sound again the next day.
Fast forward 8 years and that same horse started dragging his hind toes a little, had a little more trouble with his lead changes, a long spot that was easy for him a year or two prior was now feeling a little scary. Trainer kept pushing us to start Legend injections or to just inject his hocks. I refused and sent him for a workup with a lameness specialist. Virtually no arthritis, x-rays looked almost identical to the ones done 8 years prior. Specialist concluded that he was having degenerative changes in his hind suspensories. We opted to retire him and start him on Equioxx.