The key missing information here seems to be how many of these satellites would be required to have constant coverage of likely trajectories. This depends on the distance at which the laser remains effective. There would be no atmospheric scattering, but beam collimation is never perfect. It also depends on how fast the satellite can fire a new shot, as any warhead will be surrounded by decoys and other penetration aids. If this requires a large number of satellites, I am very sceptical. While Starlink has shown the possibility of creating large constellations, these sats would surely be much larger and more expensive. Really, Starlink makes me think something like BRILLIANT PEBBLES (https://en.wikipedia.org/wiki/Brilliant_Pebbles) would be a more reasonable alternative.
Also, could an adversary surround the warhead with absorbent chaff as a countermeasure? Or simply an ablative shield, the warhead needs one anyway to get through the atmosphere.
Still, a very interesting read from a very interesting CEO.