So during TOS, we saw that starships had different mission patches. The thing was that they weren't meant to, William Ware Theiss decided in the second season to start doing it. You can see in the first season episode "Court Martial" a crew from another ship was wearing the "Delta" or "Chevron". If you look at the series "Enterprise", Starfleet uses a Delta/Chevron in its logo, even Starfleet vessels in TOS have the yellow Chevron on the sides and on the nacelles.
So My theory is that Starfleet in general used the Delta/Chevron. (Though Admirals and Commodores generally wore a starburst patch)
I think the Delta was used in Starfleet in general, just that different ships had different patches. So why does the Enterprise use it? Well just look back to the first exploration star vessel Earth had. It was called Enterprise, so its mission patch would be that of Starfleet.
Take all this how you will, but I always found the theory or explanation that Kirks 5 year mission was so successful that they decided to have Starfleet use the insignia as stupid. There's evidence all over that they were already using it, even if you only use TOS as the only source of evidence.
(Original Delta Image from Memory Alpha. Modified by Hallgarth)