Genuinely interested, I know the basics of the torrent protocol, and I don't understand why the torrent protocol wouldn't work for streaming... I mean, you would just need to request the packets in order instead of randomly.
It would be less efficient, sure, but it would work.
Downloaders are only incentivized to give back data until they are done. Seeders are not really incentivized at all, so they can go away at any moment.
So if everyone downloads sequentially and seeders go away, then you can end up getting stuck in a situation where everyone has the beginning of a file but nobody has the last parts.
When you're streaming the protocol is not robust.
With random order you have a robust protocol that just degrades in throughput if people are selfish.
And with a webpage-bsed service being selfish is as easy as closing a tab.
It's proof that seeders seed for the sake of seeding for the entire system to work.
Update: It's also noteworthy that BitTorrent Inc.'s official torrent client (as well as the largest player by marketshare), uTorrent, offers sequential downloading, as well as selective file downloading. And the BitTorrent network remains very healthy.
Seeders are not abundant in all swarms. And nothing in the protocol guarantees their existence, and thus they do not contribute to intrinsic robustness.
Also, torrent clients that run in the background and consume few resources are hardly comparable to things that run in a browser tabs, user behavior will differ.