I can't tell you about all organizations but I can tell you about my experiences.
There is no advantage to lying about a dog's characteristics. You are going to bring the animal home and learn the truth. If the animal is not as advertised there is a greater than average chance that you are going to bring it back. This creates more of a problem than the original rescue because we planned the original rescue, we did not plan on you returning it after two weeks. People adopt a dog wanting it to work out and most people realise it could take a week or so to settle in. So it is very rare to have one returned right away. They come back weeks or months later, often with new behavior issues.
For that reason we do our best to be accurate. We are also sharing our experience; your own experience may be different. Perhaps the dog was great around the kids at the adoption events - that doesn't mean your little
monsters darlings will gel with the dog the same way. The dog may have snuggled with me because I encouraged it and is not snuggling with you because you do not.
As far as working with shelters they are hit or miss. Some shelters focus just on getting the animal out of the shelter, and others focus on saving the ones that have the best chance of being a great pet for someone. Without experience it's hard to say.