So yes, it is possible for a young Lieutenant made acting First Officer by a Captain, and later acting Captain (after relieving a former first officer who wasn't up to the task) to be left in command.
Possible for a *lieutenant*, yes. But not for a bloody cadet.
it's not totally outside the realm of possibility, especially if you're Kirk, a once in a generation hero
As I said, though...who knows if Kirk turns out that way in this timeline
as well. His entire life has been altered. He could be very different now.
I mean, look at what happened to Picard in "Tapestry". In that alternate timeline, he grew up to be a worthless 60-something Lieutenant JG with no ambition. And that was just because he didn't get into one bar fight
when he was already an adult officer! The change to Kirk's life is, arguably, even more serious, because it dates back to his own BIRTH and therefore changes his entire upbringing.