Hi Chris, I hope you won't give up on getting a rescue. It can seem like a palaver but all rescue outfits want is to get the right dog in the right home.
I too do a lot of homechecks and there's a lot goes into it and, as others have stated, a small garden with secure fencing wouldn't put me off. It all depends on the age, temperament, etc., of the particular dog you want, and what you can offer it by way of time and commitment, as well as how much exercise outside the garden. You sound like you'd be a great family.
I do the homechecks the same day or the day after the family choose the dog so he can go home asap, assuming he's already neutered, everyone's met him, a dog-to-dog's been done if they have a dog already, etc. If they already have a dog or cat or rabbit I ask to see their vaccination certificates too. I'd also rehome a dog to family that worked if they had a dogwalker or something in place - again it would depend on the dog, though it wouldn't be a young dog and it wouldn't be a working cocker.
We tried to get a dog from Many Tears 3 years ago and I must just have caught them at a bad time because I got no response after filling in the application form and no response to phone calls, so I think we just fell through the cracks somehow. So we got Alfie via the Spaniel Trust and couldn't ask for better than him.
Good luck with the search.