Yep, deadly serious there. Don't want to out myself with the full details but I am a dog behaviour consultant and the dangerous dogs act is something I know a fair bit about.
Make that accusation to the 'wrong' person, ie a police officer, a dog warden, or someone who will pass that on and YES a dog can be seized under the DDA.
The owner can then challenge the identification of the dog as a 'type' animal - they are unlikely to win this even if they have evidence up the wazoo demonstrating that the dog is the product of two perfectly legal breeds (has happened).
Or they accept the decision that the dog is 'type', then the onus is on THEM to prove the dog is not dangerous.
IF the dog is proved 'not dangerous' in a civil court case, then they get the dog back (but, this is after months and months in kennels not 'in a day or two') but the dog is now added to the exempt dog register and has to be held on a lead in public, muzzled in public (including in the owners car and in their front garden), neutered, tattooed down the inside thigh, microchipped and insured with the ONLY insurance company that insure exempt dogs. The dog can't be held in public by a minor, and technically they can be passed on to someone else, that also involves a court case to agree a new keeper, they can't just be rehomed/sold/given away.
So this means a dog who has been deemed NOT dangerous, is now treated as if they might become dangerous at any moment and live their lives imprisoned.
As an indicator of how strict those laws are, there is the famous case of Dempsey, a type dog who was muzzled and on a lead in a public place.
Her muzzle was removed as she was vomiting, police officers saw and her owner was reported, she was seized and ordered to be destroyed - the ONLY reason Dempsey survived was a legal technicality (her owner wasn't aware of the court case date) - and that took a 3 year battle.
Dempsey, a dog who had never hurt anyone, survived due to sheer luck.
The 'Breed Specific Legislation' in the UK and elsewhere is a total joke, it has been proven over and over across the world that it does NOT make the public safer, it does not reduce dog attacks or deaths.
Back to the OP - if your DC do run/squeal etc in fear, the best thing you can do is get some help in dealing with that fear.
Many dog trainers will be more than willing to help with this - a colleague of mine allows people to come to her classes where they can watch through a door, then sit in the room but behind a barrier (starting usually with the tiny puppy classes!) and progress from there, at whatever pace they feel comfortable with.
At all times the parent and phobic child have the ability to get out of the room without passing any dogs and the dogs all stay on lead (or the people pop out of the room during off lead work).
I suggest this because accidents do happen and the very thing your child fears most.. a dog chasing and jumpign at them or even nipping them - is likely to be provoked by the way your child behaves when faced with a dog.
Even very gentle, friendly dogs, can be caused to get giddy and bouncy, to chase and even nip, if a child whizzes about screaming and that situation is something many dog owners find VERY hard to teach their dogs to cope with, because its not something they come across often enough, in a predictable and manageable situation where training can happen.