Escort carriers (CVEs) did do significant service in WW2, but in specific roles. They were originally intended for anti-submarine use, for which their low speed and limited air group were adequate. Whether in convoy escort or in hunter-killer task groups, their planes could scout for prowling submarines, and often carried depth charges. If the planes didn't sink a submarine, the accompanying destroyers (DDs) or destroyer escorts (DEs) could follow up. Just as CVEs were slower and smaller but quicker to build than full sized carriers, DEs were slower and smaller, but quicker to build than DDs. DEs were also designed especially for anti-submarine use, while DDs for anti-submarine and fleet engagements as well.
In the Pacific, CVEs provided air support for island invasions, freeing the larger and faster fleet and light carriers (CVs and CVLs) from being tethered to the relevant island (group), allowing the CVs and CVLs mobility and to take on other tasks. Being 12 or so knots slower and having about a third of the air group of a full-sized CV, their role in power projection was in the support they provided and in freeing up the much more capable CVs and CVLs.
Fast-forwarding to the present, with the Wasp and America class amphibious assault ships now being capable of carrying F-35s or a mix of F-35s and AH-1Z helicopters (plus 1600+ Marines and a number of landing craft), they are kind of like CVEs, but almost quadruple the displacement and 3 or 4 knots faster (the USN has 9 of these in active commission, 2 of the newer America class being built, and a further 7 planned).