The Army does find time to train Soldiers on traditional fighting skills and cultural sensitivity.
In Sydney J Freedberg Jr., publication of The Military’s New Hybrid Warriors in the National Journal on March 14, 2009 he speaks about the Great Debate   within the Department of Defense on how best to arm and prepare the armed forces for the challenges of the war in Afghanistan and Iraq and the future nature of conflicts. Freedberg comments on how the Army is schizophrenic for spending an enormous amount of money use to recruit train and equip Soldiers to prepare for battle but struggle to find time to train their Soldiers in both traditional fighting skills and cultural sensitivity.   Freedberg statement is not accurate regarding the Army struggling to train Soldiers on traditional fighting skills.
In 2001, then a Sergeant First Class (SFC) Matt Larsen established the United States Army Combatives School at Fort. Benning, Georgia. Army units were sending their Soldiers to the course to learn techniques such as Brazilian Jiu-Jitsu, Judo, Muay Thai, Boxing and eskrima, which could be trained live and can be fully integrated into current Close Quarter Battle tactics and training methods.  
In 2002 the army adopted the Modern Army Combatives (MAC) program with the publishing of Field Manual (FM) 3-25.150, written by Matt Larsen.  
From U.S. Army FM 3-25.150 Combatives:
  * 1-1. Hand-to-hand combat — Hand-to-hand combat is an engagement, between two or more persons, with or without hand-held weapons, such as knives, sticks, or projectile weapons within the range of physical contact.
  * 1-2. Combatives — Combatives are the techniques and tactics useful to soldiers involved in hand-to-hand combat. Proficiency in Combatives is one of the fundamental building blocks for training the modern soldier.
The MAC program implemented strategies which were simple enough through repetition during daily physical training or as a warm up exercise Soldiers could be expected to...