From: Washington, D.C.
Here's a corollary question, how would a USN air wing do against the Kuznetsov? Do carriers still carry enough Harpoon / SLAM-ERs to overwhelm Kuznetsov battlegroup defenses? Or, would a battlegroup's airwing have too many JDAMs and too few stand-off mobile-target weapons.
I would guess that carrier magazine composition changes with the situation (just like VLS loadouts). If the carrier was deploying for Inherent Resolve and then got tasked to deal with a non-COIN taget, it would probably be under-loaded with standoff weapons. If the carrier was deployed to the NORLANT specifically, even just to show the flag, it would be a safe bet that the magazine composition would be significantly different.
As for weapon usage, it won't be the AGM-84s that overwhelm the defenses, it will be the ECM/Decoys/HARMs that do it. Or a Mk48.
I think a carrier that only had to focus on that Russian group could do it. If there are other threats air and sub), then you might need a second one for cover.
I think you're overestimating the capabilities of the Russian task group. The only potent surface combatant in the group is the Kirov, and the Kuznetsov has an absolute maximum of 44 fighters on board, and that's assuming they took every carrier capable aircraft they had (not likely). The fighters they do have will have very limited range or will have their operational efficiency decreased by buddy-tanking. This group might have been a tough nut to crack for the USN of the 1980s, but the Soviet tech just isn't as potent these days.
As to other threats, I don't think they're particularly restrictive to the USN in any realistic engagement. The Kuznetsov's deployment route and location are quite expeditionary, and the group will be operating a large distance from significant Russian air support. ASW operations won't really constrain a USN CVG these days, since they don't have much of an ASW mission since they lost the S-3. I suspect long-range ASW would be undertaken by P-8 squadrons forward deployed.
As for the Kuznetsov's battlegroup's role, it might be to threaten any US tomahawk shooters in the Mediterranean. Or, it will sail to Tartarus and constrain US targeting options there.
In a war scenario? Possibly. More realistically, it's an opportunity for Russia to do the following:
1. Show the flag
2. Work out operational tactics with the MiG-29Ks.
3. The CGN will also provide some additional SAM coverage off the coast (and SAM coverage has been a big Russian messaging tool).
4. Did I mention showing the flag?
I think the recent piece in USNI is a very good assessment of what the Russians are attempting to achieve with the deployment.