There's some errors, or at least misconceptions, in your timeline, which is not surprising because you are compressing several very complicated issues into this single question. To start, there's various models as to when and precisely how the Americas were peopled. There is general agreement though, that once humans crossed from NE Asia into NW North America, they dispersed rapidly, taking advantage of the "virgin" landscapes and animals. Given the diverse landscapes available, this dispersal was not necessarily uniform.

This becomes Complicated Question #1: What was the geographic distribution of human groups in the early settlement of the Americas? Unfortunately, historical demographics are exceedingly complicated, and pre-historic demographics even more so. If you have particular areas in mind, however, /r/AskAnthropology could probably help with some archaeological evidence.

What is more clear, however, is that several prominent and staple agricultural crops were first domesticated in Mesoamerica. Notably squashes, beans, and maize, the "Three Sisters" of American agriculture. Keep in mind this is a modern term though, and does not really represent the various agricultural schema throughout the Americas, but those three crops are the most visible and dominant throughout much of the continents. The question of why domestication occurred in Mesoamerica is Complicated Question #2, and again something that /r/AskAnthropology might be better at answering.

The fact of domestication of these plants in Mesoamerica meant that region could begin to support larger, sedentary societies, though this would naturally take time. Archaeological evidence for the earliest evidence for domesticated maize, for example, comes from a few cobs (which look almost nothing like modern cobs) found in what is now Oaxaca/Guerrero and date back to around 4200 BCE, meaning the actual domestication process must have started before then. Maize would continue its domestication alongside the increasing population in Mesoamerica, even as it slowly diffused outward from there, reaching what is now the Southwest U.S. around 2000 BCE, and the Eastern Woodlands perhaps as early as 500 BCE, but reaching other Eastern sites much later. The introduction to Coastal/Andean South America, in contrast, was almost instantaneous (from an archaeological perspective).

This is vastly oversimplifying things, but the development of a "backbone" crop generally tends to result in a stable, sedentary, and increasingly large population. To oversimplify further, such a population tends to result in complex, stratified, and increasingly urban societies. So it should not be a surprise that Mesoamerica was home to the earliest of such a society in North America, the Olmec polity of San Lorenzo, around 1200 BCE in what is now Tabasco (Norte Chico sites, like Caral, in Coastal Peru are the actually the first urban centers in the Americas). Why this happens would be Complicated Question #4.

So Mesoamerica already had an advantage over other regions simply by dint of being a main center of very productive crops, most notably maize. This lead to other centers of dense, urban society, alongside and following the Olmecs, in the Valley of Oaxaca, the Basin of Mexico, and, most notably for your question, the Petén lowlands and Guatemalan highlands, which were the major centers for the Classic Maya (~200-900 CE, with variances on that range depending on the interpretation and sites). This is the civilization that famously "collapsed," bringing an end to the Classic period. Why this happened, and what "collapse" actually means in this context, would be Complicated Question #5.

For the purpose of your question though, the importance of the Collapse was that the Maya population centers shifted, with lowland sites being abandoned and sites in Northern Yucatán, such as Chichen Itza and Mayapan becoming large, dominant urban centers. So from about 1200 CE to the mid-late 1400s CE, the Yucatán sites parlayed millenia of agricultural knowledge, population growth, and social organization into being a relatively more populous region than other areas. Of course, even at this time, various other parts of Central/Southern Mexico -- the Basin of Mexico, in particular -- were surpassing the Yucatán as population centers.

To quickly address the more fundamental why of your question: People moved into the Americas and settled it from the Arctic to Patagonia, and all parts in-between, including both Hispaniola and Florida. Each particular biome had its own challenges, and various groups adapted to those regions. Assuming an intentionality to settlement patterns (i.e., that past groups made a conscious decision to stop and inhabit one particular place) is anachronistic; you're assuming that any particular group had the foreknowledge of the whole of the Americas and the desire to go searching out some particular desirable region, or that any one region is universally desirable. To someone in the Yucatán, the Great Plains are barren and frigid. If you've ever been to the Upper Midwest in Spring, you'll also know that it can be just as "buggy" as any region in the tropics, with the added addition of several feet of snow in other seasons.

Assuming a why in this question, in other words, is to supplant a complex mix of inter-group relationships, ecological exploitation, socio-economic opportunity, and simple human randomness with your ideas about what particular region is "best" or most optimal. There's no single answer. Moreover, dense populations require either natural abundance (as supported hunter-gatherer/horticultural groups in California/Pacific NW) or the abundance brought by a productive suite of domesticated crops (such as was founded in Mesoamerica and diffused from there, after the Americas had already been long since peopled from stem to stern).

Finally, just to correct the most glaring error in your question, malaria is an Afro-Eurasian disease, and it (along with other mosquito-borne diseases like Yellow Fever and Dengue) was not present in the Americas until introduced in multiple events via the slave trade. Also, once introduced, malaria and yellow fever were widely present throughout even the more temperate regions of North and South America, at least seasonally. Control of Malaria inside the U.S. was actually why the CDC was founded, and they have more information, along with some nifty maps. The notion that "tropics = disease-ridden hellscapes" is, to no small extent, a holdover from European colonial ideas and biases.

I am not sure I understand your question. To my knowledge there was not "gravitation towards the Yucatan" by native groups. The reason that the Yucatan and similar regions of Central/South America were more heavily populated than North America, or deeper into South America is that food is more easily available in the Tropics than most other regions. This means that the environment can support larger populations there. It is also why historically the tropics and sub tropics of Africa, Asia, and southern Mediterranean coast were more heavily populated.