Being a country dweller, I am all for wood heaters. However, I understand your irritation, especially when you see the down side and not the benefits, but they have their place. In the city or built up areas where you need to buy your firewood, and gas is readily available I think they are a luxury and not terribly appropriate.
Before I go any further, however, wood fires are no more polluting than gas heaters in terms of CO2 produced per MJ of energy, and they are way more efficient than electrical heating from fossil fueled generators due to the efficiency losses of transferring chemical energy to heat to steam to mechanical energy to electrical energy and back to heat again. Wood heaters bypass all this and go straight from chemical energy to heat where it is needed.
The smoke you see from a wood fire is a blend of steam, incomplete combustion and fly ash. The steam is also produced by gas heaters, as is the CO2. The fly ash (which is incombustible particles too small to settle in the heater) is usually invisible in the flue gas as evidenced by the flue gas being clear if the fire has sufficient excess air for combustion. The incomplete combustion is unfortunately the down side of a wood heater. For good clean combustion, the fire needs around 100-120% excess air.
Unfortunately, 100-120% excess air means the majority of the heat from the fire is lost up the chimney. The wood heaters you buy from the shops today, have this built into them. You cannot turn down the air in a modern wood heater as much as an old one. So they are not as efficient as heaters, but superficially cleaner. The smoke from incomplete combustion results in larger particles than the fly ash from complete combustion. That's why you see the smoke. And these settle. In a country environment, this adds to humus in the soil, and so is not a bad thing. In the city, with a lot of paved surfaces, that smoke goes from visible air pollution to invisible water pollution.
Now, I said modern wood heaters are superficially cleaner. Yes, you cannot turn the air down so much, so the fire burns hotter and cleaner and produces less smoke per kilogram of wood burned, but because the excess air is higher, you actually need to burn more wood to get the same heat since more of it is wasted up the chimney.
Yet another case of green spin unfortunately, not real world benefit. In actual fact, the smoke pollution works out about the same as an old heater, but the CO2 pollution is greater, the waste heat (heat pollution) is greater, and the amount of wood you have to burn is greater. Collectively we humans are often not as smart as we think we are.
We run a wood heater, as we have no natural gas available, electricity is exorbitant and inefficient. I grow my own firewood on my property in a sustainable way. I haven't had to get wood from off our property since moving hear 10 years ago and I don't cut trees down. Ultimately, I want to change to solar and reverse cycle air con as this is the next most efficient heating after direct burning of wood, and I am getting older and don't want to be cutting and splitting and stacking and handling wood as I get older.
We collect our drinking water in tanks fed from the roof. It is sweet and clean and there is no taste or smell from our drinking water (though we have had it from bush fires and dust storms). We do not have any problem with smoke smell in our clothing from the washing line, though it is rare we run the wood heater during the day.
Al.