The key is not mass density, but unit of energy per weight. Turns out you and put more energy though fiber optic then copper for a given weight of material.
Yup totally serious. Not plan old comm fiber optic, but consider how a laser welding machine pumps kilowatts though a correctly spec'ed fiber optic cable.
Balloons get really big fast as you add payload. This means that more and more of your energy is used for station keeping rather then payload. Also with the UAV solution it is small and light enough to say fit in the trunk of a car and be handled by one person. For a balloon to carry any practical payload it is large enough where one person can no longer handle it on there own.
Main advantage of fiberoptic is two fold, first it is lighter then copper per unit of energy you can push through it. Second it is has no electrical resistance, so you don't get a huge voltage drop over long distances. What this means is you can have more payload at greater height then with a copper based electrical solution.
Cooper Bussman has made aerogel supercaps for years. You can get values up to like 100F that will fit in your hand and only weigh grams. Very cool but only 3v max so the uses are limited.
When I can buy one at mouser/digikey then maybe I'd believe you can build something neat with them in 5 years.
Cat's_Cradle is a great, easy to read, endlessly discussable sci-fi book by a master.
A diode laser, like we are using is also about 50% efficient in turning electrons into photons, so under ideal conditions you get around 25% of the electricity out the other end that you put in. I will point out that you still have to heat the Stirling engine with something like a laser so it has exactly the same lose on that end as beaming to a PV does.
Turns out that divergence is set by the wavelength, larger the wavelength the bigger the minimal divergence. Check out http://en.wikipedia.org/wiki/Diffraction-limited_system . Our lasers wavelength is 808 nanometers. Compare that to say 1 centimeter for microwave and you can see that microwaves will always require a much bigger "lens/mirror" to focus them.
Beam divergence is the biggest reason I'm aware of. Another sizable reason is kilowatt for kilowatt an infrared laser is a lot cheaper. Diode lasers (especially infrared ones) have collapsed in price in the last few years.
Time is nature's way of making sure that everything doesn't happen at once. Space is nature's way of making sure that everything doesn't happen to you.