Have you run a mineral panel on him?
Originally I thought you might be referring to something sometimes called "cold backed", which seems to be a disputed issue with some people believing it to be a physical reaction to the saddle, and some believing it to be behavioral.
Watching your videos , I've changed my mind a little, partly because I have the understanding that this is a new behavior for him, but wondering if it might be mineral related, specifically selenium. I have seen so many presentations of deficiency,from rehabbing horses, so I've stopped being surprised when it pops up as an issue for odd behaviors... it almost always seems to be the last thing people think of.
I know with our senior, as he aged, we had to increase his mineral supplement because he was no longer absorbing it the same way as he had when he was younger.
Selenium deficency is known to cause : lethargy, hypersensitivity to touch or pressure, stiffness, achyness, mood changes (among other things)
There is also a major artery? (or vein, I can never remember) which runs through the girth area. In some horses, it can cause a fainting effect if the girth is too tight, or two narrow.
I have a young mare with this issue, but she is much better in a fluffy girth (or WIDE string cinch if we're going western). She actually does something similar to your gelding, but, not nearly as gradual. (which might be that she is pretty green, and your boy has been well trained)
Do those are my thoughts, for what they are worth.