The problem with “don’t ban them but don’t let them breed more” and allow people that have one to just carry on is you just create a potentially lucrative black market for these dogs (in fact you doing that might make them more sought after), which doesn’t actually fix any problems.
Not necessarily advocating killing animals because they’re inconvenient but ultimately if they’re going to be a problem (and it certainly seems like that’s the case) then the sooner they’re banned the less harm is inflicted overall.