What a wonderful thread with tons of great responses and info.
I agree with the comments that others have stated.
One thing that has helped me tremendously is checking my own position. The upper body position for me has changed considerably over the last 6 weeks.
I can see that the others have discussed having an open chest and shoulders. It appears you might be riding in a bit of a defensive position. I had some similar issue altho probably for my own physical reason rather than what your reason might be.
The first thing I started to feel was my seat bones. How they really needed to be even in the saddle to find that comfy position.
Another huge thing for me was the hsoulders and chest. When I was able to open my shoulders and chest I swear my position really started to come together.
Are you riding with your core? Something else that I have worked on a lot.
When my chest and shoulders are open I know my core is engaged and working. Once the chest collapses then I know my abs are no longer working and if I need to halfhalt it quite possibly will come more from my hands and arms the from my core.
Another thing I have worked on is to steady the trot back with my seat and then speed it up a bit. Then steady and speed it up. This also helped my position.
The more balanced my own position is the easier it is for me to ride my horse this way. Or any horse for this matter.
As for the stiffness in your horse. I do see some "stickiness" in his hind end. There are so many things that can cause this so a vet looking at you guy just to check should help in giving you some insight. Maybe something like Adequan or legend might help if the vet thinks so.
Over all I think you two look nice and with some coaching some of the things going on here in the video will improve.