They don’t just rehome dogs to random people who turn up, the tv show is at the end of the application process when they’ve already gone through applications.
They work the same way a lot of rescues do, you apply, they vet you, then if you’re not turned down after that, they have dogs they think might suit and you meet those.
The big rescue organisations do have some blanket rules, that are usually linked to the welfare of the dogs, outside space, not being left to work etc. But they don’t actually have ones about children or other pets, they just are really really cations about saying a particular dog is suitable to live with those because if it goes wrong they’ll get fallout from it.... so they’re rare, but they do have dogs occasionally that can be rehomed with small children or cats or whatever...
Smaller rescues are more flexible, because they can be.
But even then, honestly, most people who “can’t” rescue a dog either haven’t tried, or got turned down for one dog or... there are people who just aren’t in the position to have dogs, but think they are tbh.
There’s a rescue near me where I’ve seen people posting on their Facebook that they “had” to get a puppy because they live in a flat... they rehome to people in flats all the time, but the people bitching had applied for one particular dog, who while she was small and cute was listed as needing a garden as she’s extremely reactive to other dogs, they’d applied not having a garden and planning to walk her 4 times a day in a busy city park 😐
Another common one is people not understanding why it’s not ok to work an 8 hour shift leaving a dog alone, because any home is better than being in rescue... when the dog they’re complaining about not getting was in foster anyway and an easy rehome to start with.
Or the fence issue that people complain about all the time, like, just put a fence up! They don’t care if it’s pretty stick some tall stock post fencing up, it’s not expensive in the grand scheme of keeping a dog.