So this was a bit of an oversight, but up until now I'd forgotten that you'd need a way to delete old save files.* This update fixes that. =)
* Okay, I didn't actually forget. I just figured hey, making the UI is hard and if people want to delete saves they can always do it manually. What I forgot was that if you delete a save from outside of the game, Steam's cloud-save system will actually resurrect it for you the next time you play, making your unwanted save files effectively immortal...