Offred I am sure many people will agree with you that some dog owners are irresponsible and shouldn't have dogs. I'd also advocate for more control over dog breeding and ownership for animal welfare reasons.
However, I think your logic is flawed in seeing the website as placing responsibility on 'society'. In an ideal world, all dog owners would be responsible, and would train their dogs to meet people nicely, come back when called, would not place their dogs in stressful situations etc. In an ideal world, parents/carers would educate their children about safety around animals, in the same way they educate them about the dangers of traffic, fire, water and other unavoidable hazards.
We can't control the dangers that are present in the world - how other people drive, how other people's animals behave, the presence of flammable materials or deep water. The website is equipping parents with resources so that they can make the changes that are within their power to make their children safer and their animals happier.
Your argument of 'why should the burden be on parents to teach dog safety when there are too many irresponsible dog owners' moves no-one towards a safer, happier solution - it just exacerbates the tensions which already exist between dog owners and parents, most of whom are trying to do the best they can most of the time with the limits of knowledge and education. Educating people is the way for that soemtimes inadequate 'best' to get better.