Yup, they're already having problems without even getting attacked. Shooting "down" (they wouldn't fall down) some of the satellites, could easily create enough debris to start a cascading collision effect and turn the whole orbit into a minefield.
It's even worse, because they are in orbits creating a crisscross grid, meaning debris from one satellite would cross the paths of dozens of others in a short period of time.
Also, disabling a dozen or a thousand satellites, wouldn't create a "hole" in the network over any single place, since every single satellite goes over the whole globe, replacing any disabled one.
Someone trying to attack Starlink, would either have to trigger a cascading effect, or get no effect at all.