Go back to Gene Roddenberry's personal service history and you'll see why he designed "Starfleet" the way he did.
Roddenberry was a bomber pilot with the 13th Air Force in the South Pacific during the War. The largest presence in everyone's life was the Pacific Fleet. There were thousands of ships in the big Blue Fleet (all were painted Haze Grey, btw), and the comings and goings of the gray supply ships were a constant presence in Roddenberry's life (he won the DFC, btw).
Starfleet is based on, if nothing else, the Pacific Fleet of World War II, a vast armada that as much as any instrument of war, played the most prominent role in the strangulation and reduction of the Japanese Empire. Roddenberry had a more pacific and scientific role in mind for Starfleet, but in the confident era of pre-Vietnam America that gave birth to Star Trek, Star Fleet was an armada that was fit to fight, and ready to fight.
Probably an influence, but the massive 3rd/5th Fleet concentrations of naval power don't really resemble the Starfleet we've seen. It seems more like a 19th century navy, with stations and squadrons spread all over the place, looking after national interests, backing up diplomacy, policing the colones. Especially the Royal Navy, also massive and militarily dominant, but where the "dramatic action" would be be found more in patrolling a far-flung empire than steaming back and forth in battle line exercises over and over again.