My understanding is that the worst of the pollution in Beijing is the many thousands of people heating their houses and cooking with coal.

So in this scenario, there would be a team from the government who would drive around and catch people burning coal (as that is clearly an externality) and ask them to pay them money? Similar to how the BBC drives around and demands fees if a person is caught watching TV without a licence?

The trouble with billing people for damage is it's usually too complicated to do. The BBC fee is a fixed amount to pay to see its shows. London used to have smog caused by people burning coal at home but they banned it after the 1952 smog killed 4000 people. Sadly I think the Chinese pollution has killed far more than that.