Not absurd at all. Since a billion years ago the South African (National) Defence Force has always had four equal branches:

- Army

- Navy

- Air Force

- Medical

But the USA is already mostly there: there are eight uniformed services. One is the U.S. Public Health Service Commissioned Corps that employs the surgeon general.

But you will have to stop your military's obsession with earning combat action badges. From what I can tell its creating a class-hierarchy inside the armed services.

That won't help if your aim is to create a serious non-combat uniformed service with the same prestige that mainly does nursing.