So, you're suggesting that SoundCloud seeds all the files instead of offering the download option. Perhaps this could work if "play-in-browser" vs "download" ratio is not too big.
I haven't actually kept up to date recently on the project, but the ML is super active. I believe they now have a working localforage backend to store torrents - the implementation already works flawlessly, and there is a hybrid client in the same node package family that can seed the same files to both webtorrent cleints and traditional UDP clients.
But it is extensionless and seamless and just uses webrtc data channels and websockets.
There are implementations which approximate bittorrent written entirely in javascript. It's possible.
However, I really doubt infrastructure costs dominate SoundCloud's money problems. Significant, yes. But a major engineering overhaul developing an entirely different architecture for distribution which might either not work or alienate customers is probably not an appropriate move when in that situation.
Maybe it could be something to do as a clean shut-down to help things live on (an amazing thing when dying companies do things like this to end well) but not ... expected.