Actually, the American system of strict uniformed service by functionality made us kind of the odd duck out after WWII. Integrated commands have been pretty common and they make a lot of sense in terms of war-fighting efficiency.
We’ve actually taken much the same same path now. While our various uniformed services maintain distinct training and career progression roles, operationally, troops with combat missions fall under control of independent Commands like US Central Command, which is responsible for operations in the Middle East, or the European Command.
Within that structure it’s not unusual to find staff officers from different uniformed services working together in planning and intelligence roles or even to find officers from one uniformed service commanding troops from another.
The Soviets were actually a little bit ahead of us in relying on unified commands.