I suppose you’re risking life and limb anyway by climbing, but it seems to me that retrieving ones gear is better than leaving garbage all over the place.
I hope your co-worker was OK. I’ve found that rappelling is more dangerous than climbing for a couple reasons: novices usually associate rappelling with “climbing” and are eager to do it and tend to be involved in most accidents, and experts who rappel usually do it to get out of sticky situations–which mean that they are also apt to be involved in an accident. Rappelling is one climbing technique where you are mostly dependent upon the equipment. Oh yeah, and alcohol can cause trouble too.
A friend of mine said he did once use a severed sheep shank to rappel out of necessity. They were retreating from a winter climb of Mt. Moran in Wyoming if I recall (not sure), and had one rope left. It wasn’t long enough to be doubled for the 100 ft. rappel, and they were going to need it, so they tied it off on one end, put a sheep shank in it, cut the free strand, and kept tension in the rope as the last guy rappelled. When he got to the bottom, the loose rope fell to the ground. (Hmm. You think he was telling the truth?)
I’ve demonstrated the technique–it is a real eye-opener. 'Course, my students were always belayed even on rappel. 
There’s a safer solution to that problem, RM. I thought it up after long nights pondering The Two Towers and “Nothing ever happens on the Moon”, which both contain a variant of this problem. I always felt that enchanted rope and broken bones were less than elegant solutions
What you do is the following (and I hope I can explain this without a diagram): Tie an eye of some sort near one end of the rope, and some sort of counterweight (less than your own weight) to that end (a pack would work, for instance). Loop the rope once around your attachment point, seeing to it that your counterweight is hanging over the ledge, and feed the long end through the eye you tied. This will form a loop wrapped around your attachment point, so long as the tension in your rope is greater than your counterweight, which allows you to climb down. Then, when you get to the bottom, let go of the rope and get out of the way, and your counterweight will fall and pull the long end back out of the eye. You still need to maintain tension in the rope, but if you let it go slack for a moment, you can still recover it, so long as you still have your end of the rope, and you’re less likely to lose it while cutting the central strand.
Safer is a relative term, you gotta admit.
In the wild, I’ve found that ropes are hard enough to pull down. If you try to let them pull themselves down, it never happens. One end would go half way up, and the other end would come half way down. One rule of thumb is to never let go of the end.
I did see a guy latch onto a rappel, where two ropes were knotted together, since the face was about 40 meters straight up. He wanted to use only one rope–but latched onto the wrong side (knot side) and the rope started whipping through the bolt. Someone nearby jumped over, and jammed their thumb in the eye of the bolt. Standing on a two-inch ledge with no protection, except their thumb.
ObTerm: Monkeywalk. A version of rappelling using body friction, using two ropes crossed in front of you and wrapped over behind your elbows, so you descend facing down, arms outstretched, feet wide.
By the way, I did some experimentation with severed sheepshanks here with some twine. With singly-looped ends, the knot just fell apart under tension. With doubly-looped ends, it held under tension, but then continued to hold when I released the tension. All in all, I’d have to say that, on twine, at least, that method isn’t too reliable. I have no idea how, if at all, this would scale to nylon climbing rope.
Maybe you’re cutting it in the wrong place–one of the three connecting strands should be completely loose when tension is applied. That’s the one to cut. If it falls apart before you cut…well, IMHO, you’re tying it wrong!