Just make sure you don't get sick.
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So yeah, you do have to have some timber available. But if you live in the kind of place he's talking about, there's more than enough to go around. Most of the land where I live is in crops, but there are enough trees along the creeks and in rough areas that all the people burning wood don't make a dent in them.
My parents would heat their home this way. Actually, I think they still do. They'd gather all sorts of wood from fallen trees on other peoples' land as a sort of "service" aka- they haul it away and you don't deal with it. Is it worth the cost savings? I highly doubt it. They're just not good with managing time/money.
Of course it depends on the land and the house. But here's some Reddit comments also estimating the need at < 1 acre
https://www.reddit.com/r/homestead/comments/1jnpbug/how_much...
From what I can tell both of his services are pretty popular.
And even with a well managed rotating stock of trees, you are going to at best get just over half a cord per acre. And in my area which is as close to the same weather as Northern New York as it gets, I would expect they would need atleast full 3 cords of wood to make it through a mild winter, more if it is a colder winter or if no snow builds up to help insulate or if you live on an open plot where wind can blow over your house.
I wouldn't even consider trying to survive on my own tree wood unless I had at least 10 acres to harvest off of, and it would still depend on the type of trees growing there and is still kind of straddling the edge of sustainable long term.
Maybe if you went full 16th century and started coppicing the woods and maintaining bare minimum heating you could do better, but coppiced woods also takes a decade to initial establish and maintain and nobody has coppiced woods just sitting around waiting to be utilized.