Like others have said, few rescues have the resources to perform a thorough check with every dog. To do so you would need to observe them interacting in multiple ways in multiple settings with multiple types of humans and dog.
e.g. if you want to test for children you would need babies, toddlers, girls, boys, teenagers, boisterous children, quiet children and you would need to observe the dog with all of them inside, in places they knew, outside, in places they didn't know, with toys, with food, with other dogs.
It just becomes impossible. So instead, rescues tend to talk to the last owner and question them, check some basic scenarios but then observe the dog in their care throughout time and use those observations to build up a more accurate picture of the dog.
Again, another example, at a rescue I was volunteering at recently I popped into one of their lounge rooms where a mastiff-lurcher was having time out of the kennel. I had walked the dog before with no issue. Other people had been in the room with him and described him as being a real gentleman. Something about me trying to leave that room caused him to bite me. Not hard but enough that someone else may have panicked and reacted.
That was additional information that was then used to adjust the assessment of the dog.
Also, dogs are living creatures. Their behaviours change over the course of their lifetime, regardless of where you get them from. A family dog can develop fears and reactions at any point of their life so there are no guarantees the dog you acquire is the same dog you will have for the rest of its life.