It's probably a lot more simple than that: the reason they sent Excelsior is because her captain just happened to be aboard when Kirk made a run for it. The other three or four ships in port either took longer to power up (because their chief engineer and/or primary systems were sleeping/offline) or just didn't have one of their senior officers on board at that time.
In the Trekiverse, starships generally revolve around their senior command staff, usually the Captain and First Officer and a couple of other people who are really important for some reason (varies from ship to ship). Much the same way, Enterprise-D would never pull out of port in anything short of a catastrophic emergency unless it had Picard, Riker, Data and Geordi aboard and accounted for; they could leave without Troi, or even Beverly, and a loss of Worf would definitely be felt (though tollerable for a short non-combat mission). But leave one or more of those major crewmen behind, plot logic alone suggests that somebody else who is more ready to go will take the job first.