Thank you for describing this issue, Anthony. Unfortunately I don't have an iPhone.
On my android phone, videos don't automatically go fullscreen but if I choose fullscreen, subtitles indeed do disappear.
However, when I activate the timeline, there are 3 dots on its right that open a subtitle droplist. By default, the droplist is set on "no subtitles", but it can be used to activate any available set of subtitles.
I hope this works on your iPhone too.