Heck, maybe the Guardian sent them downtime to an equivalent point in Andor's history, and it was Kirk who had to disguise himself!
I would bet this is how it happened on Kumari
in the Tears of Eridanus
-verse. Thelin and Sulu in Earth's past, with Thelin passing Sulu off as an Aenar whose ears were caught in a mechanical ice-picker.
Really, given all the times that Spock's unique abilities were the only way to save the day, it's hard to believe the ship and crew even still exist in the movie era in a Spock-less timeline. I still wish Chimes had explained how they saved Earth from V'Ger without Spock around.
I was tempted to thrown a line into The Tears of Eridanus
that Earth had been ravaged by both V'Ger and the whale probe because 1) no Spock 2) Earth wasn't important enough to rush the only ship in the quadrant to anyway.