Can you dedicate a few square feet / a square meter in the garden for a few transplants? Having clustered colonies of these tall plants has treated us well, and if you’re doing the starts yourself it’s easy enough to multisow a few cells to get a head start.
I’m a little surprised your local birds let them get to that stage, to be honest! I’ve only threshed them once, for reseeding purposes, but did so with ¼" hardware cloth, framed with 2" sides, over a wheelbarrow. The mesh catches the seeds but lets the duff fall through with a little agitation. Otherwise I’ve hung them upside down within reach of the birds we raise for them to enjoy.