You've accurately covered all the negatives with rainwater, and they are the problem. Nothing would be better in an aquarium that rainwater in a "clean" state; after all, the water in all the lakes, rivers, streams across the planet is rainwater. The parameters of the rainwater get altered by substances it runs over or through, thinking of minerals and such.
I know of aquarists who use a rainbarrel and filter the water, but these people live away from industrial areas and cities. Depending upon your area, you would have to be careful. Rainwater is naturally soft and acidic, but it's what it may pick up along the way that is the risk.