Another reason ATC might route you around an active ATCAA rather than work you through is that aircraft on maneuvers might not be on an ATC frequency and might not even be squawking a transponder signal ATC can see.
Some aircraft operate in the ATCAA on standard baro (29.
Like an MOA, a cold ATCAA can vanish into procedural non-existence.
Likewise controllers may be hesitant to have the military flight temporarily operate in a specific area of the ATCAA or reduce the altitude block until a non-participant is clear.
When a controller accommodates a non-participant through an ATCAA they certainly add to their workload and responsibilities.
The Pilot/Controller Glossary describes ATCAA as "airspace of defined vertical/lateral limits, assigned by ATC, for the purpose of providing air traffic segregation between the specified activities being conducted within the assigned airspace and other IFR air traffic.
This usually occurs when all aircraft inside the ATCAA are not on an FAA frequency and/or if they are operating in a non-standard formation when conducting such activity as air combat maneuvers.
Possible loss of separation is the number one factor in the controller deciding whether a flight will be re-routed or allowed to fly through an ATCAA.