Having worked with a dog rescue organization here in NH (of which only a very tiny percentage of the dogs rescued could be categorized as pit bull mixes), the greatest challenge we face is vetting the adopters and then educating them on responsible dog ownership. Even with mandatory classes before taking the dog, we have about a 5% return rate for various reasons, related to both the dog and the family that adopts it. The majority of those dogs can be re-homed, but some of them cannot.
The most prevalent attitude is that getting a new dog is like going to Best Buy and getting a new big screen. The new big screen is awesome, everyone loves it for a while, but then the novelty wears off. The same is true with the dog adoption, especially when the realization of the amount of work required becomes inescapable. But unlike a big screen, you can't simply turn off the dog and forget about it.
Dogs are lots of work. Raising kids (and keeping them safe) is lots of work. Combining the two more than doubles that work. Dogs must be taught appropriate behavior, but much more importantly, kids must be taught appropriate behavior around dogs and learn to recognize when a dog is not comfortable. This is what we see as the biggest challenge to successful integration into a family with young children.
For anyone with young children and a new dog at home, I highly recommend spending some time on this site - https://www.thefamilydog.com/stop-the-77/
The heartache and pain you might avoid within your own home is well worth the time spent.