If a rescue centre says any dog has to be a solitary dog it's for good reason. I've been volunteering for two Greyhound rescues for over ten years and helped rehome many other breeds through various other rescue contacts I've built up. For instance if a dog is taken into care because its attacked another dog in its home or been attacked itself you simply cannot risk placing that dog into a new home where there will be another dog if a behaviourist says it's still exhibiting aggression to dogs or fear of dogs whilst in kennels. Most good rescues, including Dogs Trust have a non-destruction policy which means many dogs with behavioural issues pile up as they need specific conditions to be met in a new place in order to be ok. If dogs were just given out you'd have a disaster on your hands. The pretty, easy going or fashionable ones fly out giving the impression that rescues are setting the bar too high. There isnt anything preposterous about it. In fact it's the opposite.
We've got Greyhounds in our kennels now that cant be homed yet as they need another dog at home for confidence reasons or in some cases they've been offered homes where there is a small dog ( some Greyhounds aren't small dog friendly). For example, I've got a two year old Brindle girl that I'm fostering in my home now that's eighteen days out of Limerick, terrified of pretty much everything except my dog. Half her tail is gone and it's clear she's had it tough. I'm going to have to be pretty selective when she's rehabilitated and ready to move on to a new home