If I'mm correct in this but CV's carry numerous squadrons of various aircraft: bombers, dive bombers, torpedo bombers, CAS bombers, fighters, etc. For the US they are the core of the fleets, the nerve centers for the admirals.
Escort Carriers: would carry only a few squadrons of torpedo bombers, bombers and and fighters. And you're right they were right they were used for escorting captial ships and convoys.
Light Carriers would have a few squadrons of bombers that would be used to cover invasion forces with bombers and CAS bombers.

With all that out of the way, here are other ideas if there is a way to trick the game in believing the CVEs and CVLs are not capitalships and don't fill up the navy capship limit. Technically, they are cheap, light units that aren't meant to be CVs or future CVBs.

Brings up another point CVBs the later evolution of CVs. The U.S.S. Midway (I believe) was an experiment with mounting heavy battleship guns on a Aircraft Carrier but it proved too unstable to launch/land planes during combat.

Ye, I already mentioned this somewhere, but you really have to contest enemy air power for invasions, and evidently CVEs where very cost effecient in that, hopefully they could be substantially cheapened and combat unworthined.

Thank you for the feedback. In response I've reduced the cost of CVEs in the most recent update (see change notes for mmore).

CVLs are categorized as capital ships currently, as they're intended in part to serve as an alternative to full blown CVs for minor nations with a lower industrial capability/port infrastructure.

CVEs are classed as screens.