I'm an idiot. Use ffmpeg copy to avoid re-encoding.

Apparently my slicing with ffmpeg before this was re-encoding
everything, which is why it was taking so long. I didn't know that this
wasn't necessary. Changing it to a straight copy avoids the re-encoding
step, which means it only takes seconds instead of minutes to slice a
game. Woohoo!
Only save if the file doesn't already exist.
Apply the same copy technique to downloading broadcast streams.
There's no good reason to re-encode everything.