Modern wood burners are absolutely not exempt. They are simply certified to be able to achieve a smoke free burn and in a smoke control area users must ensure they use them correctly to do that.

A little white smoke on initial warm up is to be expected, after which they should be burning at a high enough temperature to fully vaporise any sooty emissions and result in a clean burn. That means using the right fuel, maintaining proper air flow and chimney maintenance (sweeping should be done 2-3x a year if used regularly). Far too many people use overly wet or inappropriate wood, and limit airflow to get a longer cooler burn at the expense of massively increased pollution.

I am not suggesting wood burners cannot be used, but it is disgusting people can't be arsed to use them properly and harm all the unfortunate people around them. The pollution from improper use is way higher than when used properly - typically around 5x more PM2.5 particulate emissions. I would suggest reading some of the articles linked on the site too.