Correction, Kirk's script writers made him infallible. If he tried that logic loop with one of our robots, he'd end up a thin pink paste on the walls. Could be that "irritating jerk" filter we installed.
Except for money. Money can time travel. Forward or back.
My physics doesn't go much beyond the sub-atomic particle stage. I haven't a good grasp on how FTL can be done.
Fortunately, star-travel does not depend on FTL, just patience and persistence, like most problems.