That is definitely not what the Maya did. Or really any civilization with large permanent cities. The record of forest cutting once settlements reach a certain size is consistent and global. The Amazon simply experienced a full civilizational collapse and more than enough time untouched to recover.
It might suffice to say: The myth of sophisticated indigenous ecosystem engineers who harmonize with nature is a European trope.
Whenever any culture larger and more settled than hunter-gatherers has traditions of forest management and ecosystem balance, don't immediately attribute to them magical wisdom that Europeans somehow lack. Consider that those traditions are an ancient memory of one or more catastrophic ecosystem collapses and the practices of the people who survived them.
I mean throughout the Americas you see really complex polycultural farming traditions including the famous chinampas. In Mesoamerica you also have the milpa cycle which is a system where you clear most herbs and lightly burn it to grow your crops in polycultures. After a few years, you grow more trees and these fields turn more into managed orchards that include both fruit trees and trees grown for lumbar. Eventually you let it progress into a closed canopy system.
That's just one example, but there are plenty other I could think of that are in very stark contrast to anything you see from European civilizations.