The HMM(REIN) is a reinforced/composite squadron (O-5) who as the ACE has dets from other squadrons and is analogous to the GCE (that is a reinforced battalion) with the MEU Commander (O-6) being more akin to the Carrier Air Wing Commander who has up to eight individual squadrons + COD det assigned.
The ACE can be task organized (ie with or without Harriers) depending what is expected to be needed for pending deployment. Carrier Air Wings have done limited experiments along those lines, but are pretty much fixed in composition.
I didn't know the intricacies of how the Navy does business, but my educated guess was along these lines. Since the MAGTF is in theory very flexible, it can be tailored to exactly what is needed. The typical MEU ACE is a complete HMM plus dets from HMH, HMLA, VMA, MACG, etc etc, but there have been instances where we would deploy an HMH based ACE. The example that springs to mind for me was 1999 when a SPMAGTF did disaster relief in Puerto Rico like that.
Either that, or using the Navy CVW model just didn't sound "Expeditionary" enough for us.