Wasn't an arbitrary decision about helicopters. When helicopters were first made practical, around the Korean War, the Air Force was focused on nuclear war. Numerous generals in the Air Force never particularly wanted the close air support or tactical transport missions - generals like LeMay figured the 'real' war would be bombers and ballistic missiles, and that everything else was just support or a distraction. The Strategic Air Command's (SAC) leadership figured WWIII would be over in about 30-45 days. Given the state of the technology of the time, SAC figured it would take that long for them to drop every U.S. nuclear gravity bomb on a Soviet target. Even fighter aircraft were viewed within the context of nuclear war. Plenty of Air Force generals figured there were only two 'real' roles left for fighters: escorting bombers and air defense of the U.S. homeland.
That being said, there were generals in the USAF who recognized the value of helicopters and smaller fixed-wing transports like the CV-2 - militarized version of the de Havilland DHC-4 Caribou, capable of taking off in a mere 1,020 feet and could carry 32 troops or 2 Jeeps or equivalent cargo space and weight.
The big disagreement was about how to employ the air assets. The Army wanted assets that were assigned to Army units and responsive to the needs of those units - e.g., aircraft assigned to the 82nd flew missions in support of the 82nd, and aircraft assigned to the 101st flew missions in support of the 101st. The Air Force wanted air assets to be assigned to separate air units and employed how Air Force commanders thought best for an overall war effort - e.g., if the 82nd and the 101st both wanted missions flown then Air Force commanders would decide how to respond to those requests...so maybe the 82nd got everything it wanted and the 101st only got some of what it wanted. Or neither got all of what they wanted because the Air Force decided that the needs of the war were better filled in a different way. This last is key - the Army envisioned tactical aviation effectively functioning like mechanical cavalry, and the Air Force envisioned aircraft used as a separate part of a war effort.
This disagreement was largely created as a result of the Korean War. Close air support between the Army and Air Force was a very lengthy endeavor. The prescribed procedure was for an Army forward controller to radio the headquarters of the Army division to which he was attached, which would pass along the strike request to corps headquarters, which in turn would relay it to the Army-Air Force Joint Operations Center, which operated alongside an Air Force Tactical Air Control Center. The TACC would then order an airfield to supply the appropriate aircraft, which would contact the TACC on the way to the target and be handed off to an airborne or ground-based forward controller for actual strike coordination. It often took an hour or more between the call for the support and bombs striking targets.
Contrast that to the Navy and Marine Corps system. Due to their experiences with amphibious landings in WWII, the Marines used close air support as a substitute for artillery - artillery was often one of the last things landed, and naval gunfire support might be too risky due to the distances between friendly and enemy units. Thus, the Marine Corps system was one of forward air controllers communicating directly with pilots overhead. This system also kept aircraft loitering over the battlefield and waiting for support calls rather than on the ground (or deck of the carrier) until a call was made. The Air Force considered loitering aircraft too inefficient.
The Army, and the Air Force after it was established, had not considered close air support a substitute for artillery. This worked in the European theater during WWII, but was not a recipe for success in a fast-moving and close-quarters battle environment like Korea (or, later, Vietnam).
When the Marines entered the Korean War the Air Force wanted the air assets of 1st Marine Aircraft Wing put under Air Force control. The Marines, politely for Marines, told the Air Force to go fuck themselves. I vaguely remember reading that MacArthur himself ultimately made the call to let the 1st MAW stay with the 1st Marine Division.
Early helicopters had limited payload, range, and speed. The Air Force was not particularly interested in them...but didn't want the Army to have them. Same with small fixed-wing tactical reconnaissance aircraft and tactical support aircraft. So the Army was said "hey, y'all aren't willing to be too helpful to us, so we'll just do it ourselves" and was buying and using the things. This culminated in the Johnson-McConnell agreement of 1966 when the Army agreed to give up its fixed-wing tactical airlift aircraft and the Air Force agreed to let the Army have rotary-wing aircraft for intratheater transport and fire support.
In regards to one of your other statements - the Marines also have fighter jets. The Marines fly the F-18 and the F-35. In fact, one of the biggest albatrosses around the neck of the F-35 design is the Air Force and Navy versions see performance compromises because the 'common' fuselage means their versions have design components necessary for that stupid VSTOL fan the Marines insisted on having.