Calling out blatantly incorrect statements is not letting perfect be the enemy of the good. Electricity side effects is such a small percentage of the CO2 from a single person that neutralizing it and claiming carbon zero is ridiculous.
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.
Cut the trees down, make paper out of it, print government regulations and statistics and reports on it, store these in a vault permanently. You have a carbon sink.
It's a bit more complicated than that. The original source for the carbon that humans emit is the atmosphere itself, so overall humans are carbon neutral. And since a human body itself contains carbon, until it has completely decomposed the body itself is carbon negative.
Come on. Acting as if any "carbon positive" living is immoral, is insanity.