To credit the authors of the study, they do take some pains to bring nuance into the discussion. In particular, I am gratified to see that they preface a whole section by saying "It is widely accepted that the indigenous population of the Americas in 1492 CE did not live in a pristine landscape".

I especially appreciate that they did their research and noted that some forms of non-agricultural land-use can have large environmental impacts (like anthropogenic fire in parts of North America). I also appreciate that they don't rely on narratives of pre-colonial collapse (e.g. instead of talking about a Maya collapse impacting land use, they say " some parts of previously used cities and land were abandoned, or used at lower intensity, following widespread societal transformation in the Maya civilization".

Overall, I think they do a pretty good job of working with difficult data. As you note, population estimates are tricky at best, and so they are working with difficult data from the very beginning. Not to mention that estimates of land use are really tricky, and especially so in their methodology trying to apply flat rates of land use across huge areas of the Americas. Given all that I would be very skeptical of the results, just due to the difficulty of working with these types of data and at a scale of two continents. However, I think they generally do the best that they can given how difficult the task is.

That said, where I would criticize them most heavily is in their assumptions about disease and depopulation. As you note, they don't lean too heavily on the virgin soil hypothesis, but it is alarming how often they seriously cite Guns, Germs, and Steel as an academic resource, and also how heavily they rely on Dobyns for some of their assumptions. Dobyn's work has been roundly criticized in archaeology for far over-estimating population sizes in North America and therefore also inflating the mortality rate from introduced diseases. The way they calculate population sizes is much more moderate than Dobyns, yet they still lean on his work showing high mortality rates which is a weird mismatch. A lot of work recently has demonstrated that the major changes in population size actually followed from European contact at quite a distance, even centuries after contact. For instance, in the US Southwest, major population decline probably didn't begin until the 1620s (when first contact was in 1539). The highest losses to population also probably didn't occur until the 1700s. Part of this is that a major factor contributing to catastrophic population decline was European colonialism, including warfare and exploitation of Native labor. Disease alone wasn't enough to cause these catastrophic declines, you needed the additional factor of colonialism, which in many regions follows quite a while after initial introduction of epidemic diseases (Cameron et al. 2015). A similar time-frame is likely in Eastern North America, and even later for the Great Plains. At least for North America, major depopulation really post-dates the first onset of the Little Ice Age, which doesn't support their conclusions as well as the catastrophic decline they favor.

In summary, I think they do a pretty admirable job tackling a very difficult project. However, I think the difficulties inherent in the project itself mean we need to take the conclusions with a grain of salt regardless of how well they handled the various issues related to this project. Additionally, while they don't lean too heavily into a virgin soil hypothesis, a lot of their assumptions definitely favor a catastrophic model of population decline, while a lot of more recent work suggests that population decline was much more varied across the Americas and in some regions was actually quite a bit more gradual or at least the onset of population decline was much later than we might expect. That isn't to say that their model of population decline is entirely wrong, just that that element of the research is the least well considered.

Source:

Cameron, Catherine M., Paul Kelton, and Alan C. Swedlund, editors. 2015. Beyond Germs: Native Depopulation in North America. Amerind Studies in Anthropology.

TL;DR: Weak science, an interesting thought exercise with far too many dependent arguments to be trustworthy.

This paper is so bad that I'd avoid any historical inferences at all. I'm going to answer this from more of a science point of view as to why I wouldn't buy into this paper too much. Because I'm going to be pretty critical, let me preface this by saying I'm not denying climate change, and that I do believe CO2 is a contributing factor due to human activity. I just have a problem with a lot of the science that gets published, and this paper is a pretty shining example.

There's a big problem in general with CO2/global warming related climate modeling, which is that it rarely works predictively. With an equation modeling gravity, I can not only analyze old data, but I can measure a height of an object, drop it, and have a pretty good idea as to how fast it's going to be falling (ignoring air resistance). Sadly CO2 - temperature relationships don't perform that way.

Unlike a predictive system, global warming climate science matches a whole bunch of recorded or derived sets of data points to each other historically, but the model doesn't work going forwards (or at least, not much better than semi-random luck). Further, these models are three things: first, highly complex with lots of different inputs - solar activity, volcanic activity, albedo (how reflective the Earth is), weather patterns (the polar vortex the US is experiencing is an example), a large number of gasses (of which CO2 is only one, and not the most weighted either) - as well as coefficients and exponents for every input; second, uses measurements that are often not that accurate or precise (there end up being lots of fudge factors, like how much you adjust temperature readings depending on location, eg urban areas are warmer); and third is generally dealing with small shifts in the output (temperature).

In other words, making these arguments is kind of like building a chain. If any link is bad, the whole thing falls apart. And this paper has a lot of links in their chain. Another problem is that when you have so many variables, you can fiddle the coefficients and exponents to fit it to just about anything you want to (if you use excel a lot, take basically any small enough data set, and apply a polynomial fit, then bump the order all the way up and you'll get a fit with a 0.95 R^2 or higher). A final problem is that if you take two sets of data, especially which often have trends lasting 100-200 years and match them up, there will be statistical relationships. Whether those are truly causal, and especially to what extent they are causal is much harder to establish.

This whole issue is visible with this paper is probably best illustrated in figure 5, especially parts A and B. A is the CO2 concentration, B is the temperature shift. From around 1000-1200 AD there's a large increase in CO2 (~5ppm), but a large decrease in temperature (~0.1 C). Now, by every argument made in the paper, temperature should have increased and these results should not have happened. But the results did happen (assuming you trust the data sources, but if those are bad the whole thing is nonsense regardless). You might try and explain this with other factors, like volcanic activity (D) or solar activity (C). But volcanic activity (D) doesn't seem to be doing anything out of the ordinary, and solar activity (C) is increasing.

But OK, let's ignore that, and focus only on the times around the period in question. The temperature begins recovery around 1650, and is mostly recovered by 1750. The CO2 on the other hand is still depressed in 1750, with most of the recovery happening between 1750 and 1800.

Based on these graphs, I could propose an alternative explanation for the CO2 shifts as follows. In 1550, the continued gradual slide in temperatures made the temperature cross some critical threshold value of about -0.05 C, sparking the CO2 concentration, which had been varying pretty consistently between 2 +/- 2 pmm, to rapidly decrease to -5 ppm. At the same time, the temperatures continued to decrease to -0.15 C, possibly in a virtuous cycle (where low temperatures lead to lower temperatures). Increasing solar activity, which peaked in 1600, broke this cycle, and led to warming. Recovered temperatures which were above -0.05C around 1750 allowed for a resumption of typical CO2 behavior. Land atmospheric flux close tracks CO2 shifts, and is explained by that.

Now, the above explanation I just gave is probably (almost certainly) wrong. But it illustrates a point, which is just how easy it is to start assigning cause and effect, and explaining away events whether that explanation is good or bad. Honestly, it's the modern day scientist's equivalent to an ancient priest ascribing strikes of lightning to Zeus' ire, or rain to God's tears. It gives us the comfort of knowing why something happens, even if we're totally wrong.

I could go on. For example, Figure 6 specifically shows that all the CO2 could have come from the oceans (error bars) for the time under question. But I think that's enough...

In summary, the authors did some interesting research on population and all sorts of other things, and this is a pretty interesting review paper too. They failed to address what was happening in the rest of the world sufficiently. But most importantly for a climate argument, their science overall doesn't hold up, with far too many links in the logical chain, and far too many of those links having enough issues that I wouldn't trust it to hold any weight.

follow up question: is it true that native american populations for all points north of mexico are as low as 1 - 5 million? that seems incredibly low, like the north was almost entirely empty!