Perhaps the arrowhead became famous during the exploits of April. Maybe it was even April himself who, after being promoted, began instituting the arrowhead for starship personnel under his command. From there the arrow head could have gain wider use. By Kirk's time the arrowhead could have been used by many starship personnel while many others still used their own. By TMP all starship personnel are fully converted to the arrowhead. Then by the movies the arrowhead has been universally adopted.
As I said, VGR: "Friendship One" established that the arrowhead was the UESPA insignia 200 years before TOS, and ENT used a very small version of it on the enlisted rating patches. So it's canonical that the insignia was in use long before NCC-1701 was built, and that it was a UESPA emblem before it was a Starfleet emblem.
In my
Enterprise: Rise of the Federation novels, the approach I've taken is that the Federation Starfleet was initially a loose merger of the member worlds' space services, with each division having its own insignia. I've used the UESPA arrowhead as the Earth branch insignia, the
Constellation logo as the Andorian branch insignia, the
Exeter logo for the Alpha Centauri branch, and the
Antares logo for the Tellarite branch (with an IDIC logo for the Vulcan branch). I figure that some remnant of these original divisions is still in use by the 2260s (since Kirk says a couple of times that the
Enterprise reports to UESPA), with different insignias representing not individual ships, but different fleets/divisions operating under different administrative branches (although not necessarily broken down by member world as they were to start with). That reconciles the use of the arrowhead by multiple ships with the use of different insignias by different ships. Presumably there was some sort of administrative restructuring in the 2270s that led to the elimination of the other logos.
