@JPAU Satellites do not use propellants to stay in orbit. They quite literally fall around the world, simplified: the trick is to make the satellite fall at such an angle to ensure it never hits the Earth. It's pretty weird, and very interesting to read about if you're curious.
@JPAU orbit, as others have said, does keep things circling semi-perpetually, due to the inherent imperfections of reality all satellites eventually slow down enough to fall back to earth, so propellant would be necessary on certain orbiters (like the International Space Station) to remain in orbit continuously for decades. That is to say, like the first satellite, Sputnik, every one will fall eventually, so the GPS satellites aren't always going to be available in this manga world. So your first point is correct that eventually they will come crashing down, but it can take a really long time as propellant doesn't need to be continuously expended.