It sounds like a nice idea, but the only common factor between ATC and CCT is the certification and some fundamental core training -- everything else is super nuanced specific to each scenario. Some CCT who's specialized in deconflicting a stack overhead in wartime can't just waltz into ORD tower and say "all right boys and girls, go ahead and take a break, I've got this." Each requires domain-specific experience.

And to add on what others have said: yes CCTs represent a pool of proven ATC candidates, but depleting that pool just to knee-jerk a short term-ish solution creates an equal problem for the military -- and it's a hell of a lot harder to recruit adequate candidates for CCT. For example, they have to do like, lots of pushups...

It takes 1-3 years of facility based training in order to qualify as an ATC for a specific tower. A military enlistment is only 4 years. CCTs would end up doing their entire enlistment at one facility leaving no time to train as a CCT. You'd have to hire another person who's job is 100% CCT, at which point, why not just hire an ATC?