@CloseYourEyesAndSee there are now 20 dogs on Dogs Trust website who can live with teens and cats :-/
UK rescues are now reporting major issues with intake, several have had to start turning new intakes away because they're full. This is the pandemic puppy boom + cost of living crisis fall out.
Yes a lot of dogs do have issues in rescue, both UK and abroad, and often that's WHY they've been signed over. Dogs who are relatively issue-free and happy to live with children and other animals will be online for maybe a day, receive dozens of applications, and rehomed.
As for cat-testing, it's really difficult. For a proper cat match up they would be doing blanket swapping so both animals are used to each other's scents before they come home. Even then, any dog who is down as being "cat friendly" doesn't necessarily mean they'll be friendly to your particular cat.
A rescue's priority is the dogs in their care, not the person applying, they don't exist to provide applicants with a dog. They exist to find the, sometimes tricky, dogs in their care their ideal home and hope they don't get handed back. Does it always work out? Nope.
As someone else said, working rescue makes you utterly resent people after a while, you're lied to and shouted at on a daily basis. Have heard of applicants applying for not-child-friendly dogs, with children, being turned down. Applying again and not mentioning the children this time, as if the rescue has amnesia. The priority is always the dog's welfare, the last thing they want is a dog's behaviour to be made even worse by a poor match because someone was lying about their home.
Whether you choose to adopt from abroad or the UK, do your background reading, and make plans for if things aren't as smooth sailing as you'd hope them to be.