Depends on the definition of problem. If there reallu.was.one.or two not being responsible and all the others were picking up then even as a dog owner I wouldn't want those inconsiderate to give me and my pup.a bad name.
I had this idea years ago (requiring dog owners to provide a DNA sample, test poo, fine owners). It'll never happen in my country (UK), all the systems already in place are broken and fucked let alone adding a new licensing system. It's getting crazy over here with bully XL attacks though so maybe something will change once enough toddlers & pensioners have been mauled to death.