Why command facilities do not define a cruiser

Is command and control a defining part of the cruiser role?

Not inherently. Cruisers often became flagships, especially as their numbers declined, but command was a later added role rather than the basic definition.

A cruiser's defining functions differ from jobs that navies later assigned to scarce large hulls. Cruisers frequently served as task-force flagships because they had useful space and status, especially as their numbers fell. The United States Navy and Royal Navy both reinforced that command employment over time. Command-and-control capability can therefore characterize a particular cruiser generation without being what makes every cruiser a cruiser.