The idea isn't to remove the radar entirely but make do with something not much better than what the Nimitz class has. Without launch tubes with SM-3s no need to track things out past the atmosphere.
And we do precisely that.
https://en.wikipedia.org/wiki/AN/SPY-6
> AN/SPY-6(V)1: Also known as the Air and Missile Defense Radar (AMDR).[21] It is 4-sided phased array radar, each with 37 RMAs... planned for the Flight III Arleigh Burke-class destroyers.
> AN/SPY-6(V)2: Also known as the Enterprise Air Surveillance Radar (EASR).[23] Rotating and scaled-down version with 9 RMAs estimated to have the same sensitivity as AN/SPY-1D(V) while being significantly smaller.
Same tech, just fewer modules.