Personally, I would happily pay more for someone to come and take away my maybe-garbage and maybe-recycling, and sort through it. If the issue is purely economic - that there's a very real cost that's not accounted for - let's just make it explicit.

FWIW I think the reason why it's not that way already for recycling is to encourage people to recycle when they can. But it's clear that, at this point, the processing issue is more important than that. Then those who recycle because they want to feel good about it, can pay for the privilege. And same for donations.