Any support ship and fighter screen activity still need to be centered on a base, whether that's a land base or a carrier. Such activity patterns will inevitably reveal the location of the base. If a sub encounters an ASW, then it's a pretty good bet that there's a fleet nearby. All the Chinese have to do then is to sweep the area with their satellites. Even if the satellites are down, they can send in a squad of stealth fighters. All it takes is a position fix on the CVBG and the AShBMs can come hammering down.
What you just described is basically how I played my fleet command scenario in the SCS Fleet Command thread--but that thread was set in 2020-2023 lol. As the Chinese side, I conserved my two CVBGs until I was certain Kadena and Naha were down, then sent them after the JMSDF. I kept the USN at bay with a sub + UAV grid deployed in 100nm intervals, and if I spotted anything that resembled a carrier I immediately called in J-20s running recon loadouts to spot for DF-21Ds or other J-20s armed with YJ-12 strike missiles.
There were four issues I ran across: first, per-sortie attrition rates amongst the J-20 recon and strike teams was atrocious. Second, DF-21Ds are a one-shot weapon; ammo stockpiles for them aren't infinite. Third, that UAV/sub grid gets blown up pretty fast by the USN; those poor bastards in the subs are literally there to die so I know where to aim the PLAAF and 2nd Artillery. All of those issues compounded with the fourth problem, which was that often I'd find out that 'carrier' I sank was actually a cargo ship or oil tanker.
Any maneuver that tries to keep up the stealth of the CVBG will necessarily come at the expense of attack power. Fighters will need to make shorter and more frequent runs to compensate for the CVBG running around all over the map. And what happens if the Chinese jam the area so that the fighter pilots can't receive their rendez-vous code?
This doesn't mean it's easy for the USN to hide: it's tough for them, too. My gaming buddy who played as the US side is going to Northwestern on a NROTC scholarship and graduating into the Office of Naval Intelligence this June, so I assume he knows what he's doing. He had to run AWACS out at distant vectors and shuttle flight stations at the outer limits of combat ranges, to make it look like the carrier was somewhere it wasn't; doing that cut down on his sortie rate (by a lot) and also cut down on the loiter time of his CAP, and it made ASW hell for him because he had to sweep the "right" areas of ocean while sweeping enough of the "wrong" areas to keep me guessing. Add on top of this the 'fighter sweeps' of J-11s and J-16s I could throw at him to wear down his AWACS/ASW helos/CAP, and it was a very grim battle indeed.
The likeliest outcomes were that multiple USN ships would be sunk (including possibly one, two, or even three carriers), the CAP would be depleted, J-20s would be depleted, most Chinese OTH radar sites would be gone, and both sides would call it a day and go home. If the US side was lucky, they could nail an airbase or two, but that was usually exceptionally difficult. OTH radars were much easier to hit because those sites are stationary, very very 'loud' from an electromagnetic perspective, and fragile.
Overally, the C4ISR strain in a high-intensity naval shootout is enormous. Even in the greatly simplified C4ISR model of SCS Fleet Command, my buddy and I had difficulty controlling all our assets properly. I think the PLAN/PLAAF will need to do some serious training to actually get all their tools to work together in the 'right' way. The current quantity and quality of exercises that they're doing is simply not enough to cut it, IMO.