I don't think it's a class thing, though newspapers like to report it as such.
The difficultly with rehoming adult dogs when you have children is that many will have been mistreated badly (though plenty have just been given up through no fault of their own, owner has died/become too unwell/elderly to care for them), and that unknown factor can be a risk many people are understandably unwilling to take.
Both of mine are rescues. One was born at the home because the mum was dumped there. So I suppose she doesn't really count as a rescue proper. We originally hoped to rehome the mum but the reserve list for her was already full.
Our other dog was young, somewhere between 12-18 weeks when we rehomed her. She'd been abandoned in a local park. We've had her just under a year and she has been hard work at times. Even such a young dog can be damaged by mistreated, as she was. She's a lovely happy thing now but it's taken months and she still isn't quite right in the way she interacts with other dogs, but getting there. I won't lie and say there weren't times in the early days after rehoming her when I seriously thought about taking her back, even if they were only fleeting moments.
All dogs need a massive level of commitment but often with rescues they need more of everything, patience, time, training, love. You need to be willing and able to give them that. People who can are just as likely to be working class as they are middle class.