Time symmetry is often explained by the example of orbiting objects... What I can't find an explanation for is the moment when an object enters into orbit around another object. That clearly breaks time symmetry, since once object is in orbit (and you reverse time), it will never leave of it's own. Does this mean laws of gravitational motion are not time symmetric? Or is there some other explanation (e.g. entropy of the system)?
If we restrict ourselves to Newtonian gravity, then it is indeed temporally symmetric.
Orbits require the orbiting body to be gravitational bound to the central object---i.e. they must have negative energy (i.e. the magnitude of the potential energy is greater than the kinetic) relative to the central body. One way gravity can do this is by exchanging energy with a third body. This is still time-symmetric. (Thanks @MichaelBrown for reminding me).
In general, for something to 'enter' an orbit without such a three-body type interaction, there needs to be a mechanism of dissipating its initial (non-negative) energy. In practice, how this would happen depends on the context. For stellar systems, it could happen from tidal dissipation; for satellites, I guess it could be atmospheric drag; or for a spaceship it could be its engines.