I've finally found it, the ammunition that cost almost nothing, flies straight (over short distances) and leaves no trace. It is...
ice cubes.
I've found these to be very strong, to fly well at short range (after fifty yards they seem to get pretty squirrely because of their low density) and to be very easy to find: before they melt, that is.
I tried slinging them at a large poplar tree from about 15 yards as hard as I could: they just bounced off like rocks do.
My backstop, which is a large piece of hanging chickenwire, didn't shred them like I thought it would.
The problems come with distance throwing. I couldn't get them more than 100 yards, and I can normally throw 150-200 on my good throws.
Hopefully this is as new as I think it is, and there isn't someone else out there who's laughing at my "new discovery"