Clean Air Act came about for a reason.
Log burners are fine, like diesels, in sparsely populated areas where the smoke can quickly be dispersed. They have no place in towns and cities. Nobody is saying that nobody should have them but that having one when there are more appropriate methods is just fashion and nothing else.
As an analogy there is a house 100m or so down the street from me, whenever it has its stove on you can smell it. Fine in small doses but if everyone was doing it there would be issues.
As for how electricity is generated, in the main we seem to be using the decay heat of uranium fission and hydro power up here mainly, care to guess how much pollution that generates? Most of the country seems to be on gas now that the big coal stations are being shut down but even in a coal station you get economies of scale meaning it’s far more efficient than your small scale burner and that’s before you even get into lost heat and using back boilers to run a more efficient central heating system (radiation alone is not efficient hence the stove top fans).