The only two ways I can see to make ASAT logistics work against large constellations like Starlink is if they figure out laser-based ASAT or an in-orbit attack mechanism that shoots multiple cheap missiles/bullets (avoiding expensive separate launch per target). Neither sounds impossible, though.
Yeah it's difficult to envision a truly cost effective ASAT measure, "shooting" bullets is deceptive in that it'd still require carrying the large amount of fuel needed to effectively change orbit (especially for changing inclination) and would be a bit too messy in terms of spread of the resulting debris.
Another interesting and clean approach might be jamming the satellites from orbit. You can't go after all the satellites, but since you can predict which ones will be over when you're doing something important, you can launch vehicles to approach specifically those and jam them at close range.
