We rescued a dog from Romania and the criteria was much less strict
Well, isn't that nice. In order to get a dog (because you are, of course, entitled to have exactly what you want to make your life perfect) not only did you bring one from abroad (I hate this practice) they also had lower standards so that you could be accepted. Aw, what a heart-warming story 
I do home checks for a dog rescue and they do not accept people who work out of the house, end of story. They have no way to 'test' if the dog will be OK on its own and many dogs are not. Even if they look OK, that's not enough, dogs are pack animals and they need their pack around them.
I too have had a dog who slept LITERALLY ALL DAY when I was home. Aw, so he sleeps all day?
Nope, as soon as I stepped foot outside the house he would cry and howl. He could not be left alone for ten minutes. He was another who followed us round the house.
They all show stress in different ways, so 'not barking' does not necessarily = all fine then. It just means 'not barking'.
I'm sure some rescues do put dogs to sleep, ours does not, but so what? There are too many dogs born, pts is actually sometimes the kindest option.
And no, 'any home' is not better than no home. Dogs don't have that thought process. They don't think "well, here I am on my own, I am sad, scared and lonely, but hey, it's better than that kennel I was in last week and I'm sure someone will arrive home from their terribly important job soon so I can have company".
Nope, the only bit they feel is the now - the sad, scared and lonely part.
I'm sure some rescue will let you have a dog, they are not all terribly scrupulous about who they accept. Because, you know, ££££