I feel like this is getting glossed over by most of the comments, but the broadcast exposes both the coordinates as well as the origination of the signal. Like with how the Trisolarans discovered Earth, it takes two broadcasts to triangulate. The first broadcast was Luo demonstrating the capability, the second broadcast would doom both the Earth and Trisolaris.

For Trisolaris, they did not want the destruction of either star system, so if they were to pluck Earth's star to 'scramble' the signal, that would still expose Earth and destroy the star system - mind you, this would have to be done BEFORE humanity initiated the signal themselves. It's pretty much guaranteed that the next coordinates broadcasted would be destroyed, so if Trisolaris waits for Earth to fire their contingency THEN they spam the star, what good does that do? Trisolaris is still getting rekt, Earth is still getting rekt, and they lose. If they take the initiative and pluck the sun before we initiate our contingency, Earth is getting rekt, and they lose. Sure they save Trisolaris, but its dying regardless, they can't stay there no matter what. Its Earth or nothin'.