The time machine would use gravity so it can arrive at the same relative position in a different time. But that only gets you on the planet, not the same position on the planet.
How would rotational movement be accounted for in a generic way?
If the machine has the ability to jump in time and space (needed to land in the same spot) then it would be a real waste to use it only on earth.
So we would need a mathematical model of the entire universe that is accurate to the metre over all time. This N body is much harder than the 3 body problem.
Or we use gravity and other methods to track relative stationary position.
So, what other methods can be used to determine rotation?
So all travels to new planets require 2 jumps. One to get near and measure the rotation, and one to actually land. I'd prefer something more efficient.
Also there is the assumption that the surface rotational velocity is constant over time. This is only true for solid planets without gas or liquid centers. Insides can spin at different speeds than the surface.
When developing. You solve one problem a time. So you don't spend time trying to calculate rotation from gravity.
The beauty of a time machine on earth. Is that we can imagine most of it, and add the time travel part with our imagination to make it "plausible"
You want to add instant teleportation to any point in space at any time, while knowing everything about every planet and star including their location. Instantly.
I have a magical spell that already does everything you want. It's called "goannoyium-someoneelsium-fuckwad".