Not all time travel scenarios leave you with a time machine. The machine might itself be trapped in a closed time-like loop, could have a fixed end-point.
A Connecticut Yankee in King Arthur's Court is a classic example.
It would actually be a time-space machine if you didn't plan on suffocating.
The Sun is the center of the Universe. Interesting.
Everywhere is the center of the Universe.
That doesn't make much sense. There's no absolute "space" coordinates. Why would a time machine pick some random frame of reference instead of yours? I suppose you're correct, in a tautological sense.