It would pay to be the first nation with a Brilliant Pebbles installation, since such an installation could be used to deny any other nation the ability to create the same. Brilliant Pebbles isn't merely an anti-missile system; it could also be used to intercept rockets launching satellites, including the launch other BP systems. It's a winner-takes-all game.
If the strategic interests of America are not your own interests, all of this obviously seems very grim. If you think strategic defense is just a waste of money, then all of this is more of the same. But if you're the sort of person who thinks that billions of dollars spent on nuclear submarines is money well spent, then using Starship to launch a Brilliant Pebbles constellation is a no-brainer.