The responsibilities of all head coaches usually are spelled out in their contract, including management criteria and a moral's clause.
A head coach must manage his program, everything from budget to overseeing personnel.
A head coach oversees recruiting.
A head coach promotes the program via media, boosters, public appearances, etc., and helps increase ticket sales and donations.
A head coach oversees his/her players and makes every effort to see they are maintaining eligibility (passing grades), are good citizens (staying out of trouble), and meeting all other NCAA standards. He/she also provides moral support and guidance for the players.
A good head coach does this. Marvin Menzies was very good at this. Neil McCarthy wasn't. But what about Paul Weir? He was only at NMSU for one year as head coach, but he appears to have a "win at any cost" philosophy (as evidenced by his JC signings) which can provide short-term results, but doesn't always work out long term.