Probably the biggest threat to our carriers is large torpedoes, many of them. The problem with an ICBM kind of attack with conventional warheads is that the warhead when reentering is blinded, the "blackout" phenomenon. The shielding gets very hot and cuts of any radio emissions in or out. So, it can be redirected, and it can't use radar to ponpoint the carrier group. By the time it comes out of blackout, a carrier group, being alerted to the attack, would be ten miles away from any possible point of impact. This wouldn't require any Aegis screening of course. You could saturate the area and maybe get lucky, but even that is improbable.
A carrier is a tough kill, the battle group would have of course a number of other vessels around it including a 688 class submarine. The Soviet era Backfire bomber was designed to kill carrier groups, and the F-14 was designed to shoot down Backfires at extreme range. That era is gone. The Chinese no doubt have looked carefully at all of this, their most obvious approach is to build their own carriers, smaller ones, but plausibly capable in time. Ours are spread over the globe, or being refurbished, or in quals, so it's not easy for us to put two of them together in one theater for a period of time. I think we have ten now, and three are usually in dock, 3 are in quals, and 4 may be on station, that can be surged for a while.