Effectively you want to turn arbitrary link submissions into a podcast feed of their multimedia (probably rehosted). Then you could reuse a normal podcast client to configure when items are automatically downloaded to your phone, do playback, maintain played-status, etc. Podcasts aren't necessarily audio-only - there are video podcasts and video-capable podcast clients out there.

See https://huffduffer.com/ for an (audio) example of the web service part.