On the other side of the coin, a carrier battle group can kick a** and take names just about anywhere on Earth. Additionally the US uses combined forces when it sorties.
Definately, keep your powdor dry.
They would not need to go nuclear to take out a carrier, nor
use anything so crude as a bomber.
Six Sunburn supersonic antiship missile strikes with
conventional warheads are enough to sink a carrier.
China has already accepted delivery of these from Russia.
The Yakhont supersonic antiship missile is superior to
the Sunburn and on its way to the Chinese arsenal.
I don't think the US will go nuclear first. Using conventional
warheads, the Chinese can accomplish their first objective
without giving America a nuclear excuse.
And if that isn't enough to give you pause, think
about supercavitating torpedoes, of which we
have none.