The problem is that all light sources are to some degree divergent. You can accelerate a package to a relativistic speed over a distance of the order of 100AU,

The problem is that all light sources are to some degree divergent. You can accelerate a package to a relativistic speed over a distance of the order of 100AU, which is thousands of times less than the distance to the closest star. Consequently, the light beam that you send spreads out thousands of times less over this distance. It’s still a lot, but you have a fighting chance.