Carriers will be highly vulnerable to any advanced opponent I think. They are huge and can be tracked from space. I doubt a nation like, say, Russia doesn't know 100% of the time where US carriers are. China likely can do the same. Certainly a nation could spot any carrier which came into range of aircraft weapons deployment to its shores. Carriers aren't that fast anyway. They can't evade that quickly. Imagine you fire a Tsirkon salvo or equivalent at it from underwater position 1000km away at Mach 9 (assuming public specs are accurate). Good luck. The carrier group has no way to either evade or counter this threat unless it is in a body of water which is highly patrolled with ASW assets. Can you even do it against a weapon with 1000km range? Carriers are mostly for use against non advanced opponents.
At best it can be used for deterrent I think. There would be a high threshold where a nation would avoid sinking a carrier to prevent a casus belli even though they are so vulnerable.
If you read Western news it makes one think the US can still fight WWII in the Pacific with carriers against China, or something like that, but it makes no sense in a modern scenario.
The Type 003 will be, I think, used to provide not just air cover but longer distance strike power against other surface targets. Already with the Type 001 we saw the J-15 being tested with anti-ship missiles. So the doctrine will, I think, be a lot different to either Russian doctrine or US doctrine.
The above thinking largely reflects the prevailing school of thoughts in Russia, which certainly has its merits.
Carriers are not much a threat to Russia because of Russia's geography and its less/non- dependency on sea routes for trade, before you even think about Russia's strong anti-ship capability.
Carriers are not or at least increasingly less a threat to Chinese shores and homeland because of China's anti-shipping, air force and IAD systems.
Outside Russia and China, the US carriers are still significant threat to most other countries' seashores and homeland. One or a few CBGs can overmatch the target country's entire navy and air force.
But carriers are fundamentally are power projection platform. They project your power and presence far away from your homeland. In the middle of ocean, they're the most powerful force. In that sense, carriers are still threat to China's SLOC and trade routes, even if they're not or less a threat to Chinese homeland. And China is world's largest trading nation and will continue to trade significantly with the rest of the world, while Russia may or may not need to. For this reason, China will continue to develop its carriers to project power and presence, and to protect SLOC.
But the emergence of long-range and powerful anti-ship weapons and platforms do pose challenges to carriers as never before. As a result, some rethinking to the carriers and their escort ships are needed. Maybe a more balanced and distributed force structure between carriers and escort ships reflecting some of the advances in hypersonic missiles and other strike weapons, with less emphasis on carriers. We'll have to see.