The other problem with the scenario is you have the Enterprise stuck at Earth in 1968 for X number of episodes with no place to go and investigate. They're cut off from Starfleet and any repair facilities. They can't go to Vulcan or anywhere else without contaminating timelines.
Presumably they can leave 1968, screw up timelines and then have everything return to normal next season when they figure out how to send him back. Might even have an episode where he meets his great great great great etc... grandson who turns out to be Harry Mudd!