AEGIS can detect the missiles the Chinese have but only under cretin conditions.
What are “cretin conditions”?
Plus even “if” (and that’s a big if) we did detect the missiles our counter measures are not that great. AEGIS is old, and the Chinese by now most likely know how to operate around it.
Why do you make this assumption?
Plus the Chinese have more missiles to fire then we have to defend ourselves with. If China fired 15 missiles from different areas we probably would counter some, but many would end up hitting the carrier.
The layered defenses of a carrier battle group are designed to detect and defeat far more than 15 missiles.