These are not true hypersonic cruise missiles (e.g. a SCRAMJet), instead they are ballistic missiles with an unpowered hypersonic glider warhead. Thus, they aren't too dissimilar from the way ICBMs work for missile defense with EKVs being needed to intercept. The intercept speed of EKVs IIRC is over Mach 25.
Thus, the interception goal would be to get them in their boost phase, or mid-course phase, rather than the terminal phase after they've launched the HGV (hypersonic glide vehicle). HGV phase, I think only laser/particle weapons would work given the time between detection and firing solution. But for mid-course, they could use SM-3/SM-6. The SM-3 is specifically designed for targets like the DF. So AEGIS systems with SM-3 might be effective in protecting US carriers as long as the launches are detected. The problem is, the HGV only spends a much shorter time in ballistic phase, and once it re-enters the atmosphere, it is no longer on a predictable ballistic trajectory.
That said, military tech often works much better in theory than in reality. The USSR/Russia had a fearsome military tech on paper, the evidence in Ukraine seems to be it was hyperbole.
