If nothing else, that significantly changes the backend scale required. You should trivially be able to host (show, timestamp, text) in a SQLite database for a very long time. Expanding to screen caps adds all sorts of complexity: legal rights, consistent resizing, CDNs, EXIF stripping, what if some joker uploads shots of pornography, etc
All of these things (except the legal part) are trivial. You don't need a cdn, just use s3 or blob storage and put the blob id in the cdn. If you really want, you can use a cdn for responsive images based on screen width.
I really like that idea! That sort of skirts IP issues while giving the user better intuitive experience of the remembrance of the cinematic moment :)
PS Hate to be boogie, but I can see this as being a great driver for sales of the episodes/series themselves, i.e. "to watch the whole scene, purchase here"
I personally would love a site like this where you can also search for cinematographically beautiful frames/scenes.