First, the US Army and its NATO allies (the UK included), have been waging poor counter-insurgency for the last eight years. Not all bad in every area, but yes generally trending downward.

Cultural awareness does have a relationship to maneuver and fires and intelligence. They are all related: you must have intelligence to maneuver on the enemy, or to be effective at it, and fires can fix the enemy to allow you to maneuver. Now, not necessarily "cultural awareness", but whatever term describes all the tasks like psyops, reconstruction, political relationships, cultural attaches, language ability, information operations, civil action projects, training missions; all those soft skills fall in categories distinct from fire, maneuver and intelligence. Bad counter-insurgents cover up their lacking cultural awareness with an over reliance on firepower, like the US in Afghanistan and Vietnam previously. We have only in the last few years begun to remedy wholeheartedly our mistakes, but it cannot be done overnight.