I doubt it weighs enough to do significant damage unless storm propelled. But weigh the comfort of birds versus the cost of roof repair, screen porch, and window repair.

I'd maybe cut it back some (the higher branches) and grow a vine on it. Dead trees make wonderful vine supports!!

Maybe clematis? what other flowering vines wouldn’t be invasive?

Our native honeysuckle, Lonicera sempervirens. Hummingbirds love it!

Or any annual vines.

The issue with this...is that for permanent vines, like the honeysuckle, about the time they get established the tree will fall. However, the honeysuckel just might contain it :)