Honestly-these videos and the whole story remind me of my old guy. With him, I believe it is a hip/stifle issue, and I attribute his to a birth deformity(Cow hocked to the point that the vet at the time recommended putting him down-breeder used PVC pipe and straightened him out enough that you would never know) and arthritis after all these years. I have not had the vet per se, but over the years as this has progressed have talked about it many times with them. Some of the things he does-just so you can try and see-he does not like to go down hills (I have to zig zag if we HAVE to go down one). Altho he used to win every command class around-he lost his right lead a couple of years ago. It is obviously uncomfortable for him. The biggest issue is picking up his rear feet. You can tell it bothers him, and he is awkward about it. After the farrier he is literally 3 legged for a couple of minutes. Same for when he gets up from laying down. Totally stiff. He is improved on Cortaflex w/HLA, AND MSM. Both. He is also improved living outdoors 24/7. It keeps him moving. As much as I "feel bad" when it rains, snows, etc. He really IS happier and more comfortable.
Hope some of this helps-I have no idea how old your horse is, mine started with this in his late teens, and is now 22. He is retired, other than the occasional walking trail ride.