If the CPC believes carriers to be obsolete in 2022, it would not be building one. The fact that 003 exists means the CPC does not believe this to be the case.All governments especially superpowers in a cold war care about prestige.
Prestige means confidence from allies, and when there isn't any hot fighting, its all about posturing to discourage the other side from military adventurism. The other stuff you mention about say COVID or HSR is just about fulfilling the government's duty to provide for their popular base, not about international prestige.
As has been outlined many times, cutting edge weapons platforms largely make carriers obsolete. That will be true for Chinese supercarriers too once USA has developed similar capability.
I don't see China changing their whole doctrine to use CVs just like America. As for protecting trade routes, ofc China can get mileage out of CVs there too, but that's fairly low intensity.
Of course China cares about face. So does every other government. But not to the extent that it would waste considerable resources on a 'largely obsolete' asset.