The US Navy and US Marine Corps maintain their own TSF squadrons. Looking at the squadrons listed, it appears that there has been some expansion of the various squadrons...
Anyhow, even in this world, the USMC maintains aviation squadrons; Marine Attack Squadrons operate the AV-8B Harrier II Plus and Marine Fighter Attack Squadrons operate the F/A-18 Hornet. They're holding out on the F-35B to replace both.
During the Cold War, practically every Navy aircraft had a Marine Squadron that also flew it.
The reason the
Marines have their own air assets is because their experiences in Korea showed them that they can't rely on the Army or the Air Force for support; contrary to the Air Force, which believes the aircraft to be the ends to the means, the Marines believe their aircraft to be essentially rifles with wings. The mission is always the support of the ground combat element.
tl;dr: Marines have their own aircraft and TSFs to support their ground forces, as they can't always wait for the Army or the Air Force.