The doctrinal connection to seizure of airfield or any other piece of real estate did not exist until post 1980 and survey prior to 1980 focused on establishing and maintain current training drop zone surveys. Although it can be strongly argued the second C in CCT is for Air Traffic Control connecting this purpose to only remote/austere airfields is erroneous as it ignores the larger sustaining of continuous air landing or air drop of troops and materiel necessary for projected operations. In this larger operational capability purpose CCT exists to provide the direct ground communications link to aircraft, primarily cargo and troop carrier, flying overhead and or landing, taxing, and taking off.
Although "The young guys coming in these days, and over the last decade, have all gone in believing CCT is synonymous with JTAC" may be correct, the asserting "Our [TACP] primary mission is JTACery" isn't necessarily accurate either as although completion of Joint Terminal Attack Controller (JTAC) Qualification Course is a 5-level requirement the AFSC still has strong connections to performing duties in an ASOC. Further although the Air Force has an enlisted military occupation (AFSC) named Tactical Air Party, the capability Tactical Air Control Party (existing since WWII) exists primarily to provide the ground communications nets to the ASOC to get aircraft overhead army ground maneuver units and be the direct communications link to combat, primarily fighter, aircraft flying overhead.
Point being both AFSCs have significant reliance on bringing with them the radio communications equipment to be this communications link. Regardless differences on why the BA occupations exist have no relevance to the original question of what do BA do when not deployed and during periods there is no involvement in combat operations. However if the day-to-day duties during times there is no armed conflict has importance one can state Close Air Support requires a use of force need (declared and authorized combat operations) whereas continuous air landing or airdrop of troops and materiel can exist in a full range of military operations to include situations of getting humanitarian aid into an area which has happened and will continue to happen frequently.