Marine Recruitment Advantages
Terrans are going to be inherently stronger. Everywhere else in the solar system, the gravity is either way too low for normal humans or way too high for any humans. On Earth we are heavier, and therefore have to be stronger. Think John Carter of Mars, but somewhat less ridiculous. Plus great bone density compared to everyone else, making Terrans much harder to break compared to some others, especially anyone who wasn't born and raised on a planet at all. The Hegemony absolutely recruits on Earth for the Marines in this scenario because it's the place with the absolute strongest humans in existence and oodles of them, no contest.
Additionally, the urban infrastructure on Earth would make the logistical machine run more smoothly and easily; there's already transit centers and buildings and old training facilities on Earth, so the Hegemony may not have to build nearly as much on Earth to train marines as they would have to build in other places. Not to mention that the fact of a naturally available breathable atmosphere makes the cost of those buildings and of running those buildings much, much lower.
Naval Recruitment Advantages
Urban infrastructure strikes again. Also strength and greater bone density again, which actually matter a lot in a space navy setting.
Terrans are absolutely essential for the navy because it takes much more in g-forces to kill or knock out a Terran than anyone else. They would be able to fly faster and make harder turns than anyone else alive, giving them a leg up in any confrontation. In fact, I'd wager that the Hegemon's navy would keep ships manned solely by Terrans for the very purpose of outmaneuvering the competition in every which way.
Advantages For Both Branches
I can assume that Terrans, as they live on the planet the hegemony is based on, are pretty loyal to the hegemony. At least, some of them are. Probably enough, since there are so damn many of us. Additionally, the cost of getting them off planet is presumably high given your description. So most of them probably haven't been off-world. So you have a people loyal to a cause with no real experience of the universe outside of where they've lived their whole life. They're going to be great for putting down rebellions, dehumanizing everyone else, and generally sticking to the whole Hegemony thing. Amazing pro-hegemony low-key or high-key racists, basically.
The PRC pulled a somewhat similar trick with Tienanmen square; soldiers from provinces near the square were confronted by dissident citizens who looked just like them and had a culture exactly like theirs, and so they humanized the protesters instead of firing on them. Then the PRC sent in troops from much farther west in their country, and those troops had no basis for immediately humanizing the protesters. Unfortunately, they thus had fewer reservations about firing into the crowds.
Earth is over-populated even today. That's a logistical problem. The Hegemony needs a massive military. That's also a logistical problem. So you put two and two together, and you kill two birds with one stone.
Additionally, the implementation of space elevators and the like can vastly decrease the costs of reaching orbit. Not to mention that an over-populated administrative center of a planet, if it's functioning, will likely also be a trade hub of some sort. So you've already got infrastructure for going up and down. So taking advantage of that probably wouldn't be too hard.
Best of luck with your world. Long live the Hegemon! :)
Edit: I shortened this answer to reflect the new question, reformatted a bit, rewrote small pieces to sound better, added a better explanation for urban infrastructure underneath Marine Recruitment, and added a bit about why Terran toughness matters a lot more than I originally thought out for the Navy.
Grammar too. Always grammar.