"I don’t know much about the army but I think there’s a middle ground between fighting and paper pushing?"
Not really - you are severely career limited if you are not fully deployable (whatever the reason). Yes you might rotate through desk jobs, but it'll always be back to fully operational. And senior desk jobs will not be assigned to individuals without recent operational and command experience.
"I'm sure there’s a lot of tactical/strategic stuff that requires a combination of planning, field visits"
Yes, those are the desk jobs mentioned above.
"meeting servicemen at all levels etc"
Yes, there is a welfare structure, but that's pretty localised and usually a fairly junior role. Other visits tend to be by senior commanders - ie those in operational jobs, who will have spent much of their careers in operational jobs.