For what this website defines as history (roughly, "the story of humanity"), there's rather a lot of "pre-Columbian history" in South America, and you can't really depict it all in a single map. Population density maps for 6500BCE, 3500BCE, and 1491 AD would all look quite different from each other. For the purposes of the rest of the answer, I'm going to assume you are interested in something closer to 1491AD.
The pre-Columbian Americas have been criminally understudied at this level of scope and detail IMHO. Of course there isn't much of a pre-Columbian written record, so a certain amount of that is understandable. That being said, Colin McEvedy and Richard Jones did an amazing worldwide study of historical human population history, which included the following map in their section on The Americas:
This image is from McEvedy and Jones' Atlas of World Population History. This book is out of print and almost impossible to find for purchase, but if you can, do it. Online at archive.org.
This may not go down to the level of detail you like on who those people were, but I have seen immediate pre-Columbian language maps of South America that could perhaps be reconciled with this. Couldn't find one on a really quick search, but I know I've seen them online.
Calibrating it against the North America map you posted, it doesn't have nearly as much detail, and appears to be missing a fair bit. In particular, these days I'd expect to see a higher density in the Pacific Northwest, which M&J aren't showing. Their map I believe is likely older, so less likely to include recent research. So it might be wise to consider what it shows to be a floor, not a ceiling.
For your theoretical question, McE&J postulated about 13 million souls in the Americas (north and south) at this time, and based on this map the vast majority of them would have been living in the Azetc and Incan areas. The area of North America the Cree and Inuit were inhabiting (Canada/Alaska) may look large enough to make up for their low density, but this map projection artificially inflates that area.
The modern northern states of South America, along with the tropical coast of Brazil, appeared to have some level of farming going on as well. It may not show well in the map, but I know that farming was moving up the Caribbean islands at this time as well (probably from South America), and had made it as far as parts of Cuba.