This is a topic near and dear to me - don't get me started - haha. My wife and I have been involved with greyhound rescue for let's see....29 years now. We were heavily involved in one of the groups (board members, foster family management, etc.) when we lived in MA, but since we moved down to Pennsylvania, we've just been a foster family.
One of the smartest things placement groups do is vet vet vet any prospective owner. Home visits & discussion with all adults in the household prior to adoption, meeting all of the current pets a family might have, classes for prospective owners to get to know a few dogs, information gathering on who their current veterinarian is, and even recommendation to the "greyhound-specialists" veterinarians since they have some breed-specific things to watch out for (they are very sensitive to anesthesia, for example and tolerate one group of standard pain meds way better than others). They also do a couple of monthly post-adoption visits to assess how the dog is "fitting in".
The dogs are also fostered before they are adopted, so that their personalities can be known, their ability to tolerate cats, for example, can be known, there is a whole process. Lastly, it isn't free to adopt a dog, so there is definitely a personal, emotional and financial commitment that each prospective adopter has to meet.
In MA, we were personally involved in the adopter-approval process, which required a meeting of at least 3 members of the organization, and it didn't always go well. There is nothing worse than having to tell someone they don't really meet the standard - well, except for having to take a dog back because they were unwittingly injured by bad practices of their newbie owners.
There is also nothing more rewarding than finding exactly the right family for a shall we say "difficult to place" dog. We fostered a very sweet female who stopped racing because of an unknown injury or condition, she just wouldn't run anymore. The racing kennel wanted to euthanize the dog, of course, but we took her in and after some testing, it was determined that the dog had lupus! Very unusual, but very easily treated with a single steroid pill each day. As soon as she started the meds, she became a normal dog. Because we disclosed this condition which was likely to be with the dog throughout her whole life, no one wanted to adopt her. We fostered her for over a year. We made an appeal on the early internet, and a woman appears out of the woodwork who also had lupus. She was on the same meds as the dog! They were together 7 years if I recall correctly. It was one of the better matches we made for sure.