It would be possible to have some torrent-linke sharing feature, but I the main bottle neck is that fact that in most browsers you're not allowed direct access to the file system.
Another bottleneck is that peers cannot be accessed directly through IP addresses and ports. Instead, some broker (the tracker) must send the connection offer from one peer to the other, and then send the response from the second peer to the first. This doesn't make it impossible by any means, but it does make the protocol more complicated than it is for pure BitTorrent.