I was finally able to find the issue. It seems Jetpack keeps disabling the VideoPress license, though it doesn’t display any information saying so and if I attempt to add it again, it says that it is already active on the site. When this happens, I’m unable to upload videos and all current videos become “private”. The only way I can get it to work again is to first activate it on the dev server, add the license and then go back to production and reactive the license. At that point it disables on my the dev environment and videos once again work on prod. HOWEVER, it disabled itself again.
I’ve gone through the process again, but removed jetpack from my dev environment after the fact this time. Hopefully it doesn’t break again.