FWIW, 17th Century Europe didn't have access to the technology and knowledge that we have today, that do not require any fancy devices or technology to improve our efficiency of resource usage in the hypothetical scenario discussed in this thread.
At the time, there were many incentives to deforestation, but the main ones were to procure wood as fuel, and to clear arable land for agriculture and animal husbandry. I can at least speak to these two.
It was true in the past, that wood was an unsustainable source of heat. However, with modern wood-burning stoves, even in the nordic latitudes, this is no longer true.
Sweden and Norway have done a lot of innovation in this department in the last 80 years, because it's a matter of national security for them. They've found that it's actually more sustainable, affordable, and environmentally-friendly, to use wood as the main heating source for homes, rather than oil or coal. Again, this is only true if you're using wood stoves whose construction is informed by modern (post-WWII) knowledge. But the stoves are cast-iron, their manufacture doesn't require nanotechnology, pure silicon, etc.
On the agricultural front, it's difficult to overstate how far we've come in the last 400 years. Our caloric yield per acre on the same acreage of arable land would be much higher, today, even if you were to take away the products of modern industry (fertilizers, etc) that would presumably be inaccessible in an apocalypse.
Especially given access to new world domesticated produce, like potatoes, maize, various nuts, squashes, legumes, yams, tomato, maple, rubber.