Jet engine test facilities only really make sense when you have aircraft that can deliver backup engines on deck. One after the other, we will see the test facilities for jet engines - just not at 003.
no, if you can carry a spare engine then you can test it. whether the ship has a test stand determines the extent to which the engines embarked can be serviced and overhauled onboard.
Two things should be considered in weighing whether it is worthwhile for a carrier to be given extensive onboard facilities for overhauling it’s embarked aircraft engines:
1. what is reliability and meantime between major service of the aircraft engines embarked?
2. how long is the length of deployment for which the ship is optimized and how many sorties or flight hours is expected during its deployment.
2 is hard to know for the chinese carrier, but it would not be unreasonable to suppose she is optimized for shorter deployments than US carriers
1 since 1950s, jet engine reliability and mean time between overhaul increased by 2 orders of magnitude. down to the Nimitz class American carriers had their design origins in the late 1950s. so the presence of jet test stands on american carriers may well be an artifact of its design origin in an era when engines can be expected to require overhaul every few score of flight hours, rather than every few thousand flight hours in the modern era.