Off the top of my head, low fan speed should only make the one large radiator fan go, high speed should make both fans go (and make the large fan run at high speed).
You've done well so far though. Time to pull the water pump out and have a nosey so you can rule it out. I've never seen it personally, but I have heard of impellers falling off the shaft on cheap pumps.
When you say it overheats "quickly" when you drive it, how "quickly" does it overheat?

