The NAS study seems to think kinetic interceptors can reach the booster in minutes but yes it requires a kick stage for the delta V.
How many satellites do you need to reach that kind of coverage? And how many interceptors per satellite to hit everything from one or more nuclear subs launching from the same location near a major city. How long does it take you to react without risking false positives making the system useless etc.
The problem just gets ugly fast.
The number of interceptors per satellite was unclear but this USAF captain said "composite rocket motor and micro-missile system capable of fielding a rocket pod of 100 interceptors at a cost of approximately $60,000 per micro-missile" https://www.airuniversity.af.edu/Wild-Blue-Yonder/Article-Di...