Quarky, what do you think of this idea? The rider must pedal at an certain speed or at least a narrow range of speeds. This speed would be optimized for leg pumping comfort. At higher speeds the rider will have to pedal harder to supply the necessary torque. However, no matter how fast the bike moves the rider will still pedal at the same speed.
90 rpm (pedal speed) is the oft-quoted standard.
Most magazines, pros, and weekend warriors seem to find that to be a magic number.
Me, I'm an oddball. I prefer a higher gear and lower revs. I just find it easier to put down more force a little slower than I do to spin. Although I've heard it's bad for my knees.
As a suggestion to append to this idea: Include gearing to optimize the most common (90rmp) pedal speed, and make it easy to swap out for gearing that suits different preferences. Say every 10 rpm or so.
There are problems with the chain system. Aside from "exposure to the elements", there's the fact that the chain tends to slip off of the gear it's on when you want it to stay there, and to stay on it when you want it to shift to another. Since this happens mostly or only with the rear gear set, which has 5-7 gears of rather similar size, and rarely or never the front, which has 2 or 3 of more radically different sizes, this could be handled by just reducing the rear gear set to 2 or 3 gears (or even just 1 if the problem is instead related to the different system that's used for transfers between the rear gears). But no bikes like that seem to be built; there seem to always be lots of very similar-sized rear gears if the bike has any gear-shifting at all (and the ones without it are kids' bikes).
Don't go to department stores for bikes.

Sorry, it was begging to be said. I don't mean to slight you.
Take your bike to a bike shop. Get a mechanic to adjust your derailleurs. It sounds like your rear is a little out of line.
But your suggestion of reducing external gears is sound. Internally-geard hubs are trying valiantly to make a comeback.
My mountain bike has a road crank with 2 rings up front, 53 and 34 teeth. With a micro drive cassete in the back I can go anywhere offroad and not reach a point where the pedals can't keep up on the streets.
Mine has 46 up fron and 12 in the rear. Maybe a bigger gear would help a stomper like me, but I'm selling the bike, so I won't bother.
(I want either something a little more all-mountain oriented, or a cyclocross bike. I don't race, so the XC racer, while fun, is perhaps not the best fit for me. It's a y2k Trek STP 400, if that menas anything to you.)
I'm guessing that, with a road crank, you are a cross-country type of rider. Do you race?
On a note actually related to the OP: I've seen a few electric-pedal bikes out. And they're becoming more popular.
SOme are electirc-assist, which just gives a little extra power if the rider needs it.
But I saw another one recently that the guy bought off e-Bay, and which can do 50 kph.
It looked different, so when he stopped next to my car on the street, I asked him what it was.
But all these run on batteries. None have a generator in the pedals.