Even in today's US military, each branch has their own set of pilots.
All branches of the United States Military have an aviation program.
Army pilots assist with both offensive and defensive operations.
They perform air assault in addition to transporting both cargo and personnel.
The majority of aircraft in the Army are helicopters, but there are a limited number of fixed-wing and unmanned aircraft pilots as well.
Joining the warrant officer aviation program as a helicopter pilot means you’ll be controlling some of the most advanced aircraft in the sky. National Guard aviators are among the best in the world.
All branches of the military service have aviation units. The Marine Corps has a variety of air assets it uses to help their fellow Marines on the ground.
The Coast Guard aviation community consists of approximately 800 pilots and an enlisted workforce of approximately 2,500. Together, these pilots and aircrews fly 5 types of aircraft in the Coast Guard’s inventory, representing 200 airframes dispersed among 24 Air Stations.
As a Navy Fighter Pilot, the sky is your domain. You’ll be part of an elite group of aviators who fly and fight in the world’s most lethal jets—all from the deck of an aircraft carrier.
They all work defense and offense roles, but the primary goal of the Branch isn't air based. Each Branch has their specialty and focus mostly on it, with aircraft supporting those specialties.
With an air force, the Branch can focus on the specific requirements and training necessary for regular flying missions as a priority, rather than a support role. Yes, the Air Force supports other Branches, but they have their own missions separate from them, too.
Breaking out a Branch into it's own Air Force means it gets it's own budgets, Chain of Command (set of leaders), training requirements, bases, and a whole lot of other things that can be tailored to pilots and flying missions that could be overlooked (accidentally or purposefully) by another Branch.
Also, the military is about redundancy to make sure that critical part of the system can't fail, or if they do fail don't completely block the ability to get a mission completed. Each Branch does a little (or a lot) of what the other Branches do, but that's not their specialty. And if a critical resource gets taken out, there's plenty of other resources to step in and nearly seamlessly take over for a minimum of chaos to get systems back running correctly again. If you only have one system or Branch, you have a single point of failure that can cause complete havoc. There are times when a SPOF is a good thing, but not in the military.