That would suggest that Kirk was in command of the Enterprise for at least a bit before they started their Five Year Mission that ended in 2270.  Like sometime between 2262 and 2264. (say 2263 on average)...
		
		
	 
Yes, a reasonable interpretation, from the perspective of today. However, be aware that the passage in question 
may have been written based on the premise that the 
Enterprise was 
currently in the fifth year of its ongoing mission. I'm not exactly sure what the phrase "five year mission" in the opening monologue was 
originally intended or understood to mean, nor at what point(s) any such intent or understanding might have changed. Maybe someone like 
@Harvey or 
@Maurice could provide greater insight into that?
Anyway, as for what was ultimately onscreen, Kirk said in "Amok Time" (TOS) that Spock hadn't asked for any sort of leave "in all the years" he'd known him, and had in fact 
refused them. Nine episodes later (by either production 
or original airdate order) in "Journey To Babel" (TOS), Amanda says Spock hadn't been to see her and Sarek in four years.
Earlier, in "Where No Man Has Gone Before" (TOS), Dehner said Spock had worked next to Gary Mitchell "for years" at that point, while also seeming to
 imply that Mitchell had served on Kirk's "first command"—which, as the TMoST passage clarifies, was not
 intended to refer to the 
Enterprise herself—though ultimately only saying that Kirk "asked for" Mitchell on that unspecified assignment.
These references would indeed seem to comport with a scenario where Kirk and Mitchell came aboard the 
Enterprise from their previous assignment together, at that point both meeting Spock for the first time, with Kirk meeting Pike for the 
only time per "The Menagerie" (TOS), and this would all be at least two years before WNMHGB, but no more than four years before "Babel"...
Of course, depending on the understanding and storytelling desires of a given writer, there might be wiggle room for other interpretations. For instance, speaking strictly canonically and discounting production intent never made explicit, Kirk's first command could indeed have been the 
Enterprise. Or, if not, Kirk could indeed have asked for Mitchell aboard it 
but not received him, because Mitchell had already been assigned to the 
Enterprise, where he was already working with Spock, before Kirk ever came aboard. Kirk could even have met Spock way back at the Academy, despite not having met Pike until later, and simply been unaware of any leaves Spock might or might not have taken before he became his commanding officer. And perhaps no sort of leave was required in order for Spock to visit his parents that previous time, due to whatever circumstance. Heck, Amanda could have been speaking in terms of 
Vulcan years, for that matter!
"There always are...
possibilities."
-
MMoM