My first thought about that is you'd need a lot of satellites already nearly co-planar with the ICBM's inclination and there probably aren't enough Starlinks in any given inclination to make that realistic (granting secret dV and a sporty enough TWR). Boost phase is pretty short.

This is exactly what Golden Dome proposes to do for missile defense. It improves the math of missile defense a lot compared to past proposals and it’s not completely crazy but it’s still not certain if the technical capabilities would be there even with a fully operational Starship.