On earth, energy is infinite. Space is not. They are raising a new fricking continent from the sea floor. TNG family.
On a colony world space is relatively infinite, 100s of square miles per colonist, but energy is not. They show up with a lot of deturium, and then have to figure out how to "find" more (deturium is heavy water) before their original stock piles run out.
Less fact based, I don't think a lot of humans are allowed to live on earth, since as the seat of the Federation you need a very high security clearance to stay there like the original conditions of washington dc. Land gifted by other states, and a population of mostly clerks and politicians. Earth's deed was given to the Federation. The continuing human claim to this planet is tolerated.
If your clearance is low, and you suck, and no one will vouch for you, you and 20 thousand other similar degenerates might be asked to abandon mother earth for the frontier because the air you are breathing could be used more wisely for more bolian clerks or andorian politicians.
Joe Sisko and Rene Picard were curators for historical/heritage land marks. They are not simple small business men, because there is no money or private property on Earth.
I agree that there is a conflict between apparent paradise on Earth and what motivates anyone to go out and colonize in dangerous or unpleasant conditions. I don't think there's any support for the idea that you need a security clearance to live on Earth, though.
For that matter, that's not what Washington DC was ever like. In the 1770s the land was rural, some farmed and some swamp, with a town Georgetown. Now a lot of people do office work of various sorts because that's what the jobs are, but there are also a lot of people engaged in retail sales, running hotels, restaurants, and all the trades that keep cities running. You don't need a security clearance just to be in Washington or to visit most of the government buildings, just certain ones where highly classified work is done. The most obvious sensitive buildings are not even in DC: the Pentagon, CIA, and NSA headquarters.
Yes, there's clearly more going on to keep Rene Picard and Joe Sisko operating than just free enterprise capitalism. Star Trek is careful not to be too specific, because of the likelihood of offending some of the audience and that if they really knew how to make the economics work they'd probably be in politics making it happen instead of making a TV show.
Maybe if you're not a farmer, rancher, etc. on Earth in Star Trek's time you get a 1000 square foot apartment. (For the traditional units challenged, that's a one or two bedroom - not a big apartment by any means, but it's bigger than the efficiency apartments the working class get in expensive cities now.) If you want more space you colonize. Most colonies we assume are not as unpleasant as they got for the Maquis.