That makes sense what you are saying. The real question is how the majority of the population sees the issue.
Take for example smoking bans in restaurants and public buildings. In my county this was something the politicians didn’t want to do for a very long time because they feared the backlash of the smokers. But after a very successful public petition for enacting a smoking ban they did some surveys and found out that almost 70% of the population was for such a ban.
They then enacted the ban and all the smokers where like “The restaurants are all going to die”. Then the ban came and it was just business as usual. Nothing bad happened and actually, revenue increased because more non-smokers came to the restaurants.
I don’t have statistics on how many people would want stronger regulation of dogs, and that value might vary a lot between places. Depending on the circumstances (e.g. if it happens after a particularly gruesome dog attack) stuff like that might not even need too much political capital.
For example, after a pitbull killed a toddler who was just walking down the road, the city where I live enacted compulsory leashes in all public places. There was no shitstorm against it.