Why aren't average speeds computed over distance? Average speed is computed so that average speed = total distance / total time. This means that if you go up a hill, which takes a long time, your average speed drops. This isn't regained on the downhill, since going down takes so much less time.

If you compute the average over distance instead (i.e., your speed every tenth of a mile), the uphill and downhill are equally weighted, since they have the same number of samples. This means that your average is closer to what your speed would be on flat ground.

Your speed over flat ground seems more important to me. It doesn't depend on how hilly your route is, and that way you could compare cyclists more easily. Why don't we compute the average speed over distance instead?

Clarification edit: I know that average speeds are computed with time because we care about the average speed -- it'd take me that same amount of time to travel a certain distance if I traveled my time-average speed.

I don't care about time anywhere here (minus speed) -- the definition of average speed doesn't bother me, I'm just wondering why this isn't a more widespread metric.
This is an interesting point of view. Let's unpack this a bit.  

Assume I have a ride that is 10 miles of flat, 10 miles of ascent, and 10 miles of descent.
On the flat I maintain a constant 20 miles/hour.
On the ascent I fall back to a constant 10 miles/hour.
On the descent I maintain a constant 30 miles/hour.

My average speed for this would be:
(10 miles + 10 miles + 10 miles) / (0.5 hour + 1.0 hour + .33 hour) = 16.39 miles/hour

By the proposed calculation I would have:
(100 tenths/mile * 20 miles/hour + 100 tenths/mile * 10 miles/hour + 100 tenths/mile * 30 miles/hour) / 300 tenths/mile = 20 miles/hour

If I had an average of 20 mph I should be able to complete the course in 1.5 hours.  The trouble is that the course would actually take 1.83 hours to complete at the actual average of 16.39 mph.  I know it hardly seems fair, since by all rights you did the vast majority of miles at 20+ mph.  The caveat is that you spent by far the greatest amount of time at 10 mph.