I think they should be allowed for cultural reasons but only if cut by hand like we did when I was a kid :)

How much impact does it realistically have on climate change? I would expect it to be relatively small compared to things like owning a car?

In a perfect world we would want to reduce emissions as much as possible in every facet of life, but in the real world I think we should pick battles that have the biggest impact.

People heating their homes can be very sigificant. In the UK ~15% of all its territorial GHGs come from heating with gas: actual CO2 from the home boiler flues.

CO2 from small amounts of rural home heating is probably not the big thing to be worried about, especially if local recent biomass, eg wood from forest management. But there are still nasties (PMs, biodiversity losses, etc) to be considered and that should be dealt with in due course.

On cold mornings you can see the wood smoke hanging over the town of Taos New Mexico. It's easy to see even a little bit of haze in the otherwise crystal clear air. Taos is in general a very environmentally conscious place. The KTAO radio station has been solar powered since the early 1990s. It also has a significant population of low wage tourism industry workers.