Quoting ParagonRenegade, reply 61. Everything gives off radiation, and this is actually a bad thing.
Radiation is energy. Gas giants are huge batteries of nearly unlimited energy, almost as powerful as a star. Speaking of which, we should also be able to do something with stars.
Something none of these games really do (that I can think of) is properly address the growing power requirements an advanced space empire would require. We should be building giant power arrays in close orbit to stars and gas giants and beaming the collected radiation to space stations and planets with lasers, or transferring it through quantum entangled particles around the universe instantly.
Short of adding an entirely new resource and gameplay mechanic though, we should definitely be at least creating colonies on gas worlds. A colony on the surface of a gas planet is basically going to function like a colony on a "dense gravity" or "barren" world, just with a bit more effort required to get through the thick stormy atmosphere. Or we could build floating colonies on the outer edges of the planets, using the vast oceans of fuel below to power the engines required to keep it afloat. Collecting that fuel and sending it offworld would be a big reason to actually do that.
Terraforming a gas giant into a jungle world would not be impossible, it would just require a tremendous amount of effort removing most of the atmosphere and converting what was left into the right composition. Then genetically engineering vegetation that thrives in whatever environment (gravity/temperature/sunlight) is left. It is not at all unrealistic, given enough time we are likely going to harvest all the fuel from Jupiter leaving a rocky core only a dozen or so times the size of Earth which could be easily terraformed at that point into whatever we want. This might take 10s of thousands of years, but it will probably happen eventually.
The radiation Jupiter gives off is primarily infra-red produced by the heat inside it caused by the massive pressure and friction of its fluid ocean, and that is at best difficult to utilize. The second type of 'radiation' Jupiter and other gas giants give off is in the form of their Van Allen belts, which in the case of Jupiter can kill a person within hours and is more of a navigational hazard than a potential resource. This is because it's so high-energy (per particle), and most collection devices are useless.
Like you said later, it would be much more logical to skip all that 'radiation' jazz and focus on building floating structures in the upper atmosphere; putting anything on its solid core would require you to fly, swim and tunnel through ridiculously hazardous material (Hydrogen and Helium in their gaseous, liquid and metallic solid states) at tens of millions to billions of earth-atmospheres of pressure. Considering how many other potential colonization sites there are in the universe, I can't reasonably expect that to ever happen.
And finally, 'exhausting' the Hydrogen and Helium of a gas giant on the timescale of a conventional civilization like in GalCiv is basically impossible; there's literally nonillions of tons of resources, and there will in all likelihood be multiple giants exploited at once, reducing the drain on an individual planet.
I apologize if I'm being a smart-ass and overtly dickish, but someone has to fight for realism, right?