My expertise is in the Mississippi Valley rather than China or India, so my answer is heavily slanted towards this.

North America was rather dispersed in population, but Mesoamerica was very densely populated. The parts of Mesoamerica occupied by Nahua and Mayan-language speakers was the most densly populated part of the entire world in 1491. For scale, the population of what is now Mexico did not recover its pre-Columbian population until the 1970s. More Mesoamerica at least, the "head-start idea" does not really hold water. Had it not been for Columbian pandemics Mesoamerica might have had a population similar to that of Yangtze or Ganges. They had well-developed agricultural and economic systems that facilitated rapid population growth and a climate and geography condusive to highly-productive agricultural products (maize, squash, etc.). I do not know if your book mentioned that, but that would be a large blind spot if it did not.

The Mississippi River basin is a different question. Recent archaeological evidence suggests that humans have been building monumental architecture in the basin since the last Ice Age, over 11,000 years ago! (One mound site in Louisiana has been confirmed to be nearly 11,000 years old)* One working theory, drawing from James C. Scott, is that the Mississippi Valley was too productive for a huge population. Game and foragable foods were too common and too well-managed for people to make the leap into mass agriculture. That's because diverse foraging options makes it difficult for any kind of upper class to control the food supply and thus the population. The catch is that, while diverse food populations make for physically-healthier and more egalitarian social groups, these groups tend to have a rather low population growth because population is tied more directly to natural yield and, even with the complicated and effective means deployed by Mississippi Valley communities over the millennia, these yields can only get so high. We actualy start seeing explosive population growth in the MV with the Coles Creek (~700 CE) and the Mississippian/Plaquemine (~1000) periods, when the Mississippi Valley communities began transitioning into a more intensive, maize-reliant agricultural system akin to that of Mesoamerica. With this case in mind, the "late-start" theory you mention seems more credible. If James Scott is to be believed (and I find his arguments compelling) then the reason the MV got such a late start is because its political elites were unable to force their subjects into agricutlural labor regimes that promoted large populations until later, in part due to highly effective environmental management systems and practices maintained over thousands of years.

If "civilization" is our benchmark, the MV had a headstart of thousands of years. If "elite dominion" is our benchmark, then we can say with certaint that the MV was late to the game. Even so, Mesoamerica complicates this theory because its maize-intensive agricultural systems are fairly recent (2500 BCE) compared to rice cultivation in the Yangtze (~6,000-7000 BCE), but still experienced comparable population growth.

I suppose that I haven't really answered your question, but I doubt I could have given a concrete, definitive answer. In the end, I find this a very interesting question

• James C. Scott, Against the Grain

• Charles Mann, 1491

• Christopher Morris, The Big Muddy: An Environmental History of the Mississippi and Its Peoples from Hernando de Soto to Hurricane Katrina

• Gayle J. Frtiz, “Native Farming Systems and Ecosystems in the Mississippi Valley,” in Imperfect Balance: Landscape Transformations in the Pre-Columbian Americas

(*) Brooks B. Ellwood, Sophie Warny, Rebecca A. Hackworth, Suzanne H. Ellwood, Jonathan H. Tomkin, Samuel J. Bentley, Dewitt H. Braud and Geoffrey C. Clayton, “The LSU Campus Mounds, with Construction Beginning at ∼11,000 BP, are the Oldest Known Extant Man-Made Structures in the Americas,” American Journal of Science 322. No. 6 (2022)