Streaming while downloading videos on mobile

Youtube? Vimeo? Own hosting?

You probably gonna say I'm using the wrong tool for the task, but I just love Unity. I want to use it for this. Biggest issue here is the binary size, which could be smaller. Second one is playing in old devices, which will probably be slower than needed. Anyway...

I've got many videos of my own I want to play whichever way, in fullscreen, for android and iOS. It will be a simple list of videos for playing.

If the user does nothing, videos should be downloaded on background constantly, up to a local space quota. If they do choose one video that haven't been downloaded yet, then it will ideally be streamed and downloaded at the same time.

I know how to download them (from own server) for playing when done. That's almost the perfect way, IMHO. Making the server as good as youtube worldwide is tough, but at least we have no quotas.

So I figured maybe youtube (couple options there, no idea if they allow downloading). Maybe vimeo (found nothing there). I doubt there will be a better way to stream from own servers, though... It's hard enough to do it even on a web page.

Any experts on the area? Any better ideas?