> Considering the three body problem and extrapolating I would expect star orbits in a galaxy to be completely chaotic on long time scales?

That's more or less captured by the loss cone calculation. On a very "long time scale" the loss cone will probably be occasionally replenished by stars coming in from scattering on larger scales, but the timescale for this to become significant for galaxies is substantially longer than the age of galaxies (and also longer than the Hubble time, which is the ~age of the Universe). So, at least from this back of the envelope calculation, that doesn't solve the final parsec problem.

See, e.g., https://www.astro.umd.edu/~richard/ASTRO620/Dynamics_Lec3.pd...