Actually, it was only until much later in history that people really started settling in the Andes mountains. The earliest monumental architecture and urban centers in Peru were located in the coastal river valleys descending from the mountain slopes. Caral, a significant early center from ~2600 BC is located along the Rio Supe. Later settlements did move eastward into the Andes, but stuck to river valleys. Cerro Sechin, from ~1600 BC, is further inland in the Rio Casma valley. Chavin de Huantar, the center of a pan-Andean cult beginning ~800 BC, is along the Rio Maranon. Along the Rio Santa, a valley called the Callejon de Huaylas, one can find many settlements from this period: Tumshukario, Huaricoto, etc. Modern cities follow this pattern. It's not until the decline of the Chavin culture that we see habitation anywhere away from rivers. This is around 300 BC, long after the beginnings of "civilization's" hallmarks. These settlements are highly fortified, walled towns often literally built on the edges of cliffs, home to the chiefdoms of people like the Recuay. The decline of the unified regional culture lead to plenty of warfare, so people were willing to build canals from lagoons if it meant a defensible town.

Yes I know there were the Mississippian cultures, but mounds of earth don't really compare with vast empires and huge stone structures.

Why aren't earthen pyramids like Monks Mound (circa 1100 CE), the Great Mound of Troyville (circa 500 CE), or Mound A / the Bird Mound of Poverty Point (circa 1500 BCE), to name a few, comparable to other pyramids around the world? True, they use a different building material, but that material has it own unique challenges that must be overcome in order to build with it on these scales, and often transported great distances to achieve the desired results.

As for their territorial vastness, it's often hard to detect archaeologically, but artifacts associated with Poverty Point have turned up along the Mississippi-Ohio River system from New Orleans to southern Indiana, though the majority are clustered in northeastern Louisiana and northwestern Mississippi (map). As for Cahokia, its influence seems to have extended from southern Indiana (the Angel Mound site near Evansville seems to have been the local capital for a polity that reached up the Ohio River as far as Louisville, Kentucky), while to the north it had influence on sites like Aztalan in Wisconsin and Red Wing in Minnesota. However, whether this was direct political influence or indirect cultural influence is exceptionally hard to tell; maybe they were independent polities modeling themselves after Cahokia, maybe they were tributary administration centers funneling resources back to Cahokia. There's evidence from the Mill Creek sites in southern Illinois that indicates that tributary administration centers were something in Cahokia's political arsenal.

In fact they did. If you look at the places where the earliest Mesoamerican polities originated, they are either, on swampy areas (great for wetland agriculture) or in river valleys. For example, San Lorenzo was located in swampy areas in the Gulf Coast and had a lot of chinampa agriculture. Paso de la Amada, La Blanca, Izapa and all the Soconuscan phenomenon were also located either on swampy areas or near rivers (in fact, the Soconusco area is bordered by two swampy areas). Further into Maya territories, the earliest settlements like Chiapa de Corzo was located near the Grijalba river or the site of Ceibal was located near the Pasion river. San Jose Mogote and the Zapotec society begun in the best agricultural areas in the valley of Oaxaca around the Atoyac river. Further into the coast, the earliest settlements in the Pacific coast like La Consentida or Charco Redondo are in swampy areas or near the Rio Verde river. To the west, the earliest Guerrero sites like Teopantecuanitlan or the Mezcala culture also took place in a river valley around the Balsas drainage (and in fact maize agriculture may have started here). Now, into the valley of Mexico, the earliest settlements like Tlatilco, Tlapacoya or Cuicuilco were located either on the coast of lake Xochimilco or Lake Chalco, especially well suited for wetland agriculture because those lakes where fresh water, contrary to the Texcoco lake. These are the most well known early polities in Mesoamerica, all of which were located on pristine agricultural areas. That being said, there is far more to them than just looking at their ecology but that is a different question.

I would just like to chime in and offer some West Mexican examples. The culture I study, the Tuechitlan culture, flourished from the Late Formative (~300 BC) to the Classic (550 AD). It was situated in the highland lakes region of Jalisco, primarily around Tequila volcano. There were several lakes, rivers, and marshlands around the volcano, but some of those features have since disappeared due to an increase in agriculture. There is even evidence of chinampas, but due to heavy sugar cane production and poor record keeping during colonial times, they may be historical creations rather than prehistoric.

Further south in the Postclassic (1300 AD to 1530 AD) you have the Tarascan empire situated around Lake Patzcuaro. At one point the Tarascans expanded as far north as Lake Chapala in Jalisco, but retreated in order to better protect their borders. Lake Cuitzeo is another major lake containing several settlements being excavated by a French team.

In Zacatecas the site of La Quemada from the Chalchihuites culture (~300 BC to 1200 AD) was situated near several large lakes. The other large Chalchihuites site of Altavista is near the Rio San Antonio.

And to throw out an outlier just for funsies, there is the Old Copper Complex situated around the Great Lakes that made use of copper for tools and decorative items as early as 3000 BC.

I'd also like to point out that you're making a bit of a bad comparison. The Aztec, Incan, and Mayan civilizations came much, much, much later than the river valley civilizations you're trying to compare them to.