The new wood stoves are very safe and efficient. I cringe when I see wood stoves from the past being sold on FB marketplace. These old steel boxes did ok but needed a lot of attention, especially cleaning flues and stove pipes multiple times a season.
I did a lot of research before finally settling on a Pacific Energy Super. This thing burns so well that unless you just loaded it, there's no smoke from the chimney. My neighbor has a Harmon and it's constantly smoking from the chimney. Mine has the secondary burn and it works awesome. Others utilize catalytic converters.
Of course, wood definitely plays a big part. Rule of thumb is to only burn wood that has been split, racked and exposed to airflow for at least one year.
Another thing I am particular about is the flue. I feel much safer using a good quality stainless steel double wall pipe with no bends or the least amount possible. It was additional work but I ran my pipe straight up through 2 floors and out the roof. I'm also lucky to work in heat treatment and got furnace insulation that can withstand temps in excess of 2000⁰... I used it to line my flue where it passed through the ceilings and roof.
One last thing is a good carbon monoxide / smoke alarm. When my brother and I installed a pellet stove, the first night it ran, my alarm went off. I woke up to it and a splitting headache. Come to find out there was a section of the exhaust pipe that was leaking.
All and all, wood stoves and pellet stoves are very safe with newer models and insurance companies should update their stipulations to allow the use of these heat sources with an initial inspection by a qualified person prior to approval.