I'll have to leave it to RNE to fully address your view of what American Indians believed. I would just observe that two culture that have a concept of property rights may differ greatly in their view of the extent of the rights one has in the property. The Indian family that "owned" a plot of land for many generations may still have managed it within the framework of a philosophy of stewardship.
In contrast, the concept of property advanced by American Conservatives is broad and expansive, and ideally (from a Conservative perspective) confers god-like powers. In the ultimate perversity, we consider property rights to be so strong that a property owner has the right to destroy said property. In many cases, when the effects are local, nobody really cares. But we happen to be talking about agricultural land, which transforms the question into one of the shared interests of society. Agricultural land represents our food supply. An irresponsible farmer who destroys his land destroys it for much longer than his own lifetime, depriving his heirs of value, or reducing the value of the land for whoever buys it; and overall, reduces our civilization's food supply by the amount of growing land lost. How do you balance the ideology of private property with the fundamental survival concerns of society?
Robert, it's the idea of personal ownership and responsiblity for land that encourages conservation.
Tom, it's far more complex than that. Whether land ownership leads to conservation depends a lot on the supply of new land to replace land exploited and exhausted. You yourself described the dynamic: "no need to do anything but use up resources and move on". You may have been describing nomadic peoples, but you're also describing to a "T" the traditional pattern of resource extraction by business. When it's cheaper to just use up a resource and move on, that's exactly what people will do.
The problem with relying on the freemarket to address problems in the nature of the "tragedy of the commons" is that the signal that you've used up a resource and it's time to move on comes in realtime; that is, at the very last moment, when the problem becomes impossible to deny, and you're facing a shortage of whatever it is. Sometimes that doesn't leave enough time to adapt, and we all suffer. When it comes to environmental problems, the idea is to detect the problem early enough to take action with far less pain and disruption, on our schedule, not in a panic facing nature's Malthusian deadline. Central planning is counterproductive when the concerns are mostly local, but there's no other way to address global problems than through global coordination.
Remember that Smith's free market runs on the fundamental premise that individual actors are making strictly selfish, local decisions, and have no global overview of the market. That makes the free-market system, by its own definition, the wrong tool to address global problems. Having a "mixed economy" has become a matter of survival.