Read the aviation major command board precept. A requirement for CVW command and is "significant career experience in carrier aviation operations." Note this is the same requirement for CVN command. "Significant career experience in carrier aviation operations" is defined later to be completion of a DH and/or CO tour in a squadron deploying aboard CVNs, specifically including CVW helicopter squadrons and excluding "expeditionary" VFA and VAQ squadrons.
Also not that other major commands are specifically limited to VAW/VAQ/VFA.
For comparison, the current MCMRON SEVEN commadore is a HSC pilot who had no prior experience in MIW. He has command of four ships and is expected to be the day to day MIW expert in theater. Proof that at the O-6 command level, the navy is looking for well rounded leaders.