The US Navy is structured for a blue ocean, peacetime global power projection role. It is mostly a vestige of the Cold War. The Soviet global blue ocean nuclear sub threat was considerable. It was comparable to the German WWII uboat menace, which it was modeled on. While still existant, the current Russian threat is fractional. Submarine threats of this day and age are primarily from diesels, which have less range and endurance, and are intended to keep us far away from coasts and choke points. In this regard, they're pretty effective. Countering this threat is a different animal from open ocean ASW. We require a lot of small ASW frigates. When I wrote "the US Navy does not need lots of ASW convoy escorts," I had in mind escort carriers specifically. Helicopters now provide most of the ASW air power formerly only a CVE could provide. Our issues with getting large numbers of frigates are well known, and have been discussed here. Certainly we need ASW convoy escorts, just not CVEs. They would need defending as much as our larger carriers or logistics ships, and the goal should be using lots of smaller ships offensively hunting subs closer to defended coasts, not defending larger warships trying to get near those coasts. The modern buzzword is assymetry. The defenders have large numbers of cheap weapons making the offense risk a few expensive high value ones. You don't counter that by sending forth more high value assets (like CVEs.) We need to get cheaper faster. Drones are thus the future, and we are behind even there.
Ultimately, US Navy doctrine says that the best counter-submarine weapon is another submarine. If you look into it, there is actually considerable worry going on over our poor submarine production numbers, and there are active measures being undertaken to boost production. This is actually a higher priority within our naval circles right now than a good many other programs. There are also UUVs...unmanned underwater vessels. We're lagging on those as much or more than with the USVs (unmanned surface vessels.) We would like to accelerate those programs as well.
All this costs money, requires manpower that often is lacking, and production facilities that don't exist. Will the person in possession of the magic bullet PLEASE step forward quickly and pass it over to us? Thank you!

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Why do you think the US Navy does not need 'lots of ASW convoy escorts'? The Chinese submarine fleet as well as the Russian are not small, and logistics are a significant issue (not only) in the Pacific.
ASW is not currently a strength of the USN. I think it lacks ASW ships to protect both the carriers and the convoys.
I am not sure if smaller ASW carriers would be the solution, although they were seen as such in the late stages of the Cold War, when a number of small European ASW carriers were introduced. At that time, USN supercarriers also had significant ASW forces on board, but these have long since disappeared.