No, they were letting perfect be the enemy of good.

It takes decades for those trees to grow to firewood size. In the meantime there will be carbon "in flight". Since the trees will be cut down and burned again, effectively replanting only reduces emissions by some fraction, dependent on the growth time. Burning wood, especially at your cabin, has a high emission rate, so an electric stove might be better, and fairly certainly is if powered by a solar panel.

Trees die. Decomposing, or burning, gives off all the carbon they soaked up.