What makes you think the woodys frighten off the other birds?
They are bigger birds, but are quite dumb when it comes down to it, and have every right to try and build their nest wherever they can.
Why can't people let birds do "what birds do" and enjoy it,

rather than try and intervene all the time?
Other than that, get a license and a gun and shoot them and be done with it!
Sorry you didn't want them harming?.....Bahhhhh I don't know the answer
Now where's that culling grey squirrels thread.......