Thanos Thanos:
The effectiveness of the anti-ship missiles might be highly over-rated. None of the current Russian or Chinese missiles have sunk anything of substance yet, and the USS Stark survived (with great loss) when it was hit by an Iraqi Exocet. It's fair to question the survivability of the destroyers or frigates. In the case of the carriers though, where it's been wildly claimed that they're obsolete due to the new anti-ship missiles, there is zero real evidence to suggest that they're sitting ducks. The best evidence is that there hasn't been a US carrier sunk in action since 1944 and since then their defences have only gotten stronger. If the USS Forrestal could survive it's own ordnance blowing up inside the ship after a major on-deck accident during the Vietnam war then the super-carriers right now could be considered optimistically as the actual definition of "unsinkable".
While I agree that a modern battleship could shrug off several hits from Exocets or even Silkworms, the problem is that China and Russia are building ballistic anti-ship missiles that they believe can sink carriers (whether or not they can will hopefully never be known).
Those missiles hit from high altitude and can probably punch through just about any armour you could conceivably put on a ship, just because of sheer kinetic energy, nevermind the warhead. Carriers are protected by a battlegroup with lots of SAM defences, but in the OP scenario, the battleship would sail in there by itself and lack that same SAM umbrella.
That's why I believe a battleship would need an extensive Aegis SAM capability to protect itself from the batteries of DF-21 and DF-26 launchers the Chinese have in service.
Thanos Thanos:
As for the battle ship idea, who knows really. I doubt that the US Navy has the resources right now to invest in anything like that these days, moreso after it appears the USS Gerald Ford carrier might be a botch requiring a complete rework thanks to the repeated failures of it's magnetic deployment system to launch aircraft. The idea of something the size of the old USS Missouri or the Yamato, but even more heavily armoured, just bristling with more missiles than can be counted is fairly fascinating though. Toss in a dozen or so railguns installed on it too and suddenly that thing becomes the most lethal ship ever deployed, kind of like a sea-bourne Death Star. I have no doubt the technology is there, but it would be a matter of will and financial resources to be able to get it done.
I agree that the battleship is an interesting idea, but I'm not sure if they could work alone as the OP proposed, but two or three working together might be able to do what was envisioned in that scenario, although the cost of several would probably be prohibitive, just like the Zumwalt destroyers turned out to be.