My mistake. I nonetheless doubt that Trek's fictional Earth made this enormous progress in the second half of the 21st century while its population exploded like it does nowadays.
Population growth, resource shortages and climate change are the main, interconnected problems for our species in this century.
Lets shortly delve into demographics. People propagate massively mainly for economic reasons, children can work on the fields and care for you when you are old. The first cause vanishes once a country industrializes and the ration of people employed in the primary sectors decreases sharply. The second vanishes when there is a public pension system or when lifespans expand and people privately save for old age.
In short, if everybody is moderately well off on United Earth the population should stagnate or decline.
Then again Trek has indeed implied, mainly in TOS, that humankind spreads for economic reasons. The wheat in Trouble with Tribbles, miners and so on. This became irrelevant in the 24th century due to replicators and it was never implied that there are land shortages (like when the Europeans settled the Americas). As you pointed out, there are no Starfleet ships designed for the transport of a large number of people.