I would think it was somewhat similar as it is even today.
If an aviator stays in longer than his obligated service, likely thinking of a career, he gains experience and promotion in rank. Within his own "community" (VF, VAF, VAW, HS etc) he becomes qualified in various aspects of that community's mission. Some include section leader, division leader, maintenance check pilot, NATOPS check pilot, tanker pilot, etc. Later he or she branches out into qualifications within the Air Wing, such as a strike mission leader wherein he would be the lead of a gaggle of airplanes composed of whatever the mission required and regardless of what type of plane this pilot was actually qualified in.
For many years, a CAG in particular was qualified in at least a couple of different types in the Air Wing and often would lead a strike, but squadron COs and XOs usually were qualified to lead strikes as well. The CAG can't lead them all if the theatre of ops is extended.
In any case, any Air Wing operation lead by someone other than the CAG would fly it in the type of plane he was qualified in. By "lead", that does not necessarily mean the airplane in front; some strikes can be lead by an E-2 NFO flying out ahead on a particular station, etc.
In McClusky's case, he WAS the Air Group Commander at Midway, and I surmise had been at least checked out in the SBD at some point prior to deployment, including carrier quals. It would depend on when exactly he was promoted to the group commander. If it was while deployed he might have got a quickie checkout at sea, which would be virtually impossible in today's aircraft.
Today, with the FA-18 being predominant this becomes somewhat quirky because the CAG may be an FA-18 pilot but does not necessarily have to be qualified in another airplane type. However he or she likely gets a little orientation time in the E-2 and H-60. Also, the fact that NFO's can be designated as a CAG causes some adjustments in these common practices.