"No, we don't take cushions." "Sorry, no stuffed animals." "Not if it needs repairs."
I'm not blaming them -- they no doubt get a lot of plain old garbage, and have to draw the line somewhere.
But that leaves some major challenges for people trying to get rid of stuff that isn't ready for the landfill. And that's where nonprofit garage sales can come in. If you can accept items that the established resale places can't, you'll be doing a service to both the donors and any interested buyers, and help the environment by reducing landfill volume. (If you really think no one will be interested in buying an item, put it in a "free" box.)
At day's end, of course, you may need to get creative about disposing of things, as you yourself realize that Goodwill and Salvation Army aren't going to take all the leftovers. Scout around your community ahead of time. The local animal shelter, for example, may have a use for bedding that others won't take.