I started having the same problem in the Discovery+ app on one of my Apple TV's recently. Then yesterday, my second Apple TV started showing the same issue. The interesting thing is that the HBO and Discovery+ apps are supposed to be
combined into a single app soon.
Will have to see if these same solutions work for Discovery+ because turning off the captions on each program is really annoying. Last night, I turned it off on the show I was watchiing and had to pause the video for a few minutes. When I started playing again, the captions came back on all by themselves!