Hi Mark
What versions of Safari on what platforms?
I've tried this on Safari for macOS 10.14, iOS 12.0 for iPhone and 12.0.1 for iPad and it always shows the captions and always shows the captions language picker.
The only problem I could see was that if you have more than three languages, the language picker popup that's displayed when the video is being played inline isn't tall enough to display all the languages, but presumably that's easily solved with a bit of CSS.
Worth nothing that full screen video in iOS isn't being displayed in the browser, it switches over the iOS media player - so Adapt has no control over how that looks and only minimal control over how it behaves. Not sure if the same applies to Safari for macOS but it certainly looks like its doing something similar.
Safari for macOS full screen:
(player controls showing due to me pressing keys to take screenshot)
Safari for macOS inline video:
Safari for iPhone fullscreen: