He's a puppy. Breed irrelevent - see the DEED NOT BREED campaign for further information on that. Or come and meet me and my kids, we've owned SBT x, we've fostered Staffies and we regularly handle and interact with a dozen or more Staffies and a similar number of other Bull breeds, plus various German Shepherds, Rotts, Mastiff crosses, many from unknown backgrounds. Sadly that number is ever-growing.
We help save them from dying - very often as a result of being thrown out by some bastard - and we have the difficult job of helping to rehome them despite claims from the general public that "the papers" say that Staffies are dangerous. Didn't you say that upthread too, OP?
Please bear in mind that 99.9% of newspaper journalists tend to know and have as much to do with dogs as I do journalism. This is often quickly made evident when a headline screams "Pit Bull attacks!" and beneath it is a photo of a dog which is clearly a Staffordshire Bull Terrier. I've seen that a few times now.
On the suject of Pits, and just as an aside, you say your child was chased by what you described as a Pit Bull "type". Did you know that it takes a court of law and several expert witnesses to define whether a dog is a "Pit Bull type" and even then it's not an exact science?
Please also bear in mind that SBT were originally bred for bull baiting and later for dog-fighting and not human-fighting and that there is no proven correlation between the two. The motivation is completely different. A dog with any trace of that former trait - and similarly a dog bred to fight still - will almost always be easy to handle and very biddable - they need to be in order to be picked up, moved around and treated when they come out of the fighting arena.
This poor little mite has shitty owners who let him out unsupervised - there is far, far, far more liklihood of harm coming to the puppy than there is to you or your kids. ScuttleButter's wise words are, as ever, well-informed and from a background of solid experience - it's really all about the owners.
You did the right thing by contacting the Dog Warden - well done. Do so again if the owners don't heed his/her warning please.
You may consider too asking owners with children and dogs whom you trust - a mum at the school perhaps - to help you gradually introduce your fearful child to dogs and to overcome that fear. Happy to provide tips as to how to do it - it seems to me that your DC can't avoid dogs in the street, parks, others' houses and so on so it would be beneficial to address the fear if possible rather than live with it and stress every time a dog goes past.