This is sort of what happened with Google Chromecast with Google TV. I bought that on a technicality for my parents over an Apple TV. My mom (who isn't a native English speaker) was watching another foreign language show on Netflix and whenever she paused on the Apple TV the seek bar would come in and overlay itself on the subtitles. She was frequently pausing just to catch up on the long sentences to read them and then unpausing just as quickly. This wasn't an issue on the Android-based Netflix, where the subtitles remained in view.
Well OF COURSE because it's fucking Google they started shoving more and more ads onto the device, to the extent that my parents actually get pretty confused on how to properly navigate the thing. It makes me so mad.
I have a similar situation. Have you thought about submitting feedback to Apple regarding the issue so they can take the scroll bar into consideration and subtitle placement? https://www.apple.com/feedback/
That sounds like a plan. I'd search online for advice from other users beforehand, regarding which solutions/apps are confirmed to work well with the exact device, OS version etc.