As a disclaimer, you may want to wait on someone who has experience with Mesoamerican or South American cultures as they would have had a far more socially complex hierarchy involved in the process of growing crops.

As for North America, much of what we see was extremely decentralized. Crops would have been crown by individual communities for members of the community to consume, with many households also tending a small garden as well. We can see through feasting events and other gatherings that at times elites would have been able to pull together large amount of food to feed a gathering, but we also know that elites used things like prestige goods to establish a level of control over the population. This is basically a system of bartering for goods, the elites just had access to more exclusive goods which is where the value came from.

Growing crops wouldn't have been a venture one got into to get rich, instead it was a source of reliable calories to go along with what you hunted and gathered in your local environment. We know that in some areas agriculture was done on a scale that impacted the local environment in ways like straining local water quality and destroying forests; but the scale of it was far smaller than the large scale commerical systems that would follow in the subsequent centuries.

As you mention, climatic changes would have also played a rather large role on the effectiveness of cultivation practices. Around 1000 years ago we see climatic shifts that seem to match the rise in large urban sites along the Mississippi river and in the Southwest. These were not directly caused by climatic shifts, but it does seem that the weather was more tolerable to crops in some situations.

The biggest difference between pre-columbian and European cultivation practices is scale. Cahokia may have put a strain on the local environment, but ultimately the population relying on the crops grown near the city would have been limited by the environment itself. You didn't have shipments of grain being sent to other parts of the New World. There was no incentive to over produce, infact the incentive would have been to find that sweet spot where you grew enough to survive while also not impacting the environment you relied on for the other portion of your daily caloric intake.

European style plantations would have been established with profit in mind, ideally they would be supplying product all over their respective trade networks. A plantation in Alabama may only have a few thousand people living in the area around it, but would potentially have thousands more it was supplying once those goods were shipped out. This is when we start seeing agriculture out pace the local environment, maybe you can raise some cattle near the Amazon but if you need to feed the demands of an entire region you'll eventually need to start chopping down some trees.

All cultivation impacts the local environment, but the impact is directly related to the quantity of goods you are looking to generate. A small village of 100 people will have far less of an impact than a town of 50,000; especially if that town is exporting the crops to a separate region for profit. The globalization brought about by the Columbian exchange supercharged the impact one farm could have, suddenly you could feed far more than would have normally lived in the region. This led farmers to shape the local environment to their own specific needs, and on a scale that would have a huge impact.

Both systems are ultimately difficult to sustain, but the larger the system the higher the stakes for both the population as well as the environment.