There's a thread at the moment about why landlords can't/won't accept housing benefit tenants. And this is related but isn't having a go at you, because I understand. Much as I hate it, I do get it.
I'm in a horrible situation right now, where I'm struggling to find housing. The council/government does not categorise me and DH as high need so expect us to find private landlords.
Although it may not do much, it would mean the world to me (and many others in our situation), if you could write to your local MP to tell them why you cannot pick up their slack.
I know it's a big ask, and I know you owe me (or any HB claimants) nothing. But I'm asking because you've always been wonderful to go to for advice, and because your opinions hold more weight than ours with officials. And maybe, just maybe, it might help make things better.