One thing to remember is that North America has some fairly well-defined ecoregions, and these seem to have had a measure of influence on American cultures and societies. On the Plains, for example, we see big game hunting and hunting / gathering persisting for much longer than it did east of the Mississippi. Significant differences are also evident between cultures in the Great Basin, west of the Rockies, in the Southwest, and so on. So I'm going to restrict my comments to the Southeastern US.

North American societies were well-adapted to their own environment and often displayed progressive social structures and cultural institutions

This is a bit inaccurate, actually. At least in terms of being "well adapted" to their environment. There's a mounting body of evidence that indicates indigenous eastern / southeastern American cultures, at least as far back as the Middle Archaic (ca. 9000 years ago) were making significant intentional modifications to their landscape, including the use of fire to control / modify ecosystems.

https://scholar.google.com/scholar?hl=en&as_sdt=0%2C18&q=sediment+increase+native+american+prehistoic&btnG=

There's also growing evidence that sediment loads in rivers began increasing significantly with the adoption of cultivation and agriculture. A number of freshwater mussel species, for example, declined in abundance at a time when cultivation and agriculture were increasing. It has previously been interpreted as resulting from natural fluctuations in flow regimes in waterways, but the species in question thrive in shallow riffle run, rocky-substrate environments, and with increased sedimentation (from runoff on cleared land) these habitats would have declined.

Coupled with these, it's also important to remember that structural requirements for sedentary groups would have impacted local forests / woodlands, and the cutting of timber for construction probably also denuded large areas.

So we need to be careful not to assume the classic "noble savage / in tune with nature" trope for indigenous American societies.

With respect to social structures, also, I think applying terms like "progressive" is dangerous. Much of what we know about social rank and status in Southeastern societies comes from accounts by Spanish explorers, and they portray societies with established, hereditary hierarchies with significant differences in social status, established leadership that could be considered "royalty" in its treatment, including royal families. The Spanish recognized the Southeastern Mississippian leaders as not all that dissimilar from the sorts of leadership they were used to, which says a lot.

The fact that they didn't develop similar technology with the Europeans is explained by the fact that they strayed away from centralised, agricultural states - which also explains the almost complete lack of empires (looking at you, Cahokia).

The development of metal smelting, alloying, and so on (which is usually the most critical thing that people key on when discussing the difference between North American versus Old World cultures) relied on a long historical trajectory in parts of the so-called Old World that involved interaction, trade, exchange of information, migration, and access to the necessary materials. Whether or not groups in eastern North America developed metallurgy or not could really be viewed as largely a historical peculiarity.

The most significant difference between the Americas, though (and especially North America) and the Old World is the lack of large domesticable animals. The extinction of such animals at the end of the Pleistocene in North America could be viewed as one of the most significant divergence points between the Old and New World.

Draft animals were an enormous advantage to developing the kind of centralized agricultural states that you mention. Lacking draft animals, North Americans relied on human labor. This limited them in some ways-- you can pull heavier loads / do more work by hitching up another ox to a cart or plow, but you can't really accomplish the same thing with humans.

So one of the biggest determining factors in the differences in how Old and New World urbanized cultures developed is the differential access to labor from draft animals in the Old, and the lack in the New.

As a result, agricultural methods developed differently. It's also useful to remember that domesticated grains in the New World were fairly limited-- corn, chenopodium and a few others before that-- and by contrast Old World cultures had a variety of relatively high-yielding starchy seeds (wheat, barley, etc.) and other domesticates that could be intensively cultivated and produced a lot of food.

Coupled with draft animals, it was possible in parts of the Old World to produce enormous amounts of surplus food, which can support larger and more densely-occupied settlements.

Basically, what you have is a series of historical factors-- available domesticable plants and animals, mainly-- that differed between the Old and New Worlds. While environmental determinism is appropriately eyed with suspicion, the fact remains that we do have to consider what indigenous cultures around the world had available to them.

A human powered urbanized culture will look very different from one that can be powered by draft animals.

The trick is to remember that one is not "more advanced" than the other. They're simply very different! We can talk about complexity (i.e., hierarchization, for example) but applied as most archaeologists use it these days, that's a quantitative term referring to levels of hierarchy and administration, not a qualitative one synonymous with "advanced" versus "simple" / "primitive."

What prevented the agricultural, centralised-state society from spreading in the region of North America?

To begin with, let's examine the "Agricultural" aspect of this question. At the time of European contact, the majority of of the Eastern Woodlands (the US east of the Mississippi, the neighboring southern fringe of Canada, and a bit of the Plains states west of the Mississippi) actually were agricultural societies. Many cultures in the Southwest were as well, but I'll focus on my own area for here.

Agricultural in the Eastern Woodlands goes back at least 3000 years, with the development of the Eastern Agriculture Complex (sunflowers, goosefoot [a relative of quinoa], sqaush, etc.) in the middle Mississippian and lower Ohio valleys. By 1800 years ago, maize starts showing up in the Eastern Woodlands, but it's rare. It doesn't become a staple until around 1400 years ago (again in the middle Mississippi valley), and doesn't become widespread until around 1000 years ago. It coes some dietary health problems as it replaces more nutritious but more labor-intensive EAC crops. This problems persist until beans become common in the area around 700 years ago.

This long history of agriculture is one of the defining features of the Eastern Woodlands.

With the spread of maize we also see the spread of Mississippian traditions through much of the Mississippi valley, the lower Ohio, and throughout the Southeast. These are generally regarded as more centralized than prior and contemporary tribal confederacies. We can get into a debate over how centralized they actually were, but at the very least do appear to have had a central figurehead or two, even if councils remained potent forces in their society. Was there something about the spread of maize that facilitated the raise of more centralized powers; or perhaps the other way around, with centralized powers facilitating the spread of maize? Or was it just coincidence, with other cultures emulating powerful Middle Mississippian societies for reasons unrelated to the spread of maize? Certainly there were plenty of societies, particularly in the northeast, that adopted maize without adopting all the other trappings of Mississippian society, so obviously maize along isn't a sufficient reason to adopt a more centralized political structure.

The era of Mississippian-style centralization (to the extent that it actually existed in the first place) doesn't last long. Cherokee oral history records their rebellion against their priestly overlords. Archaeology at the Angel Site in southern Indiana shows a rather formal end to their Mississippian-style rule; following an earthquake, the temple atop the major mound burned and one last layer of earth added to seal the site. Afterward, the people of the Angel Site moved slightly downriver to become the Caborn-Welborn culture and lost the more distinctively Mississippian traits of their culture. No more moundbuilding, for example, and more egalitarian community planning.

The Mississippian powers in the southeast continued into the contact era, but even they were starting to lose some of their traditional Mississippian trappings. When the Apalachee moved their capital from Lake Jackson in the mid-1400s, they didn't build any mounds at the new locations (the capital bounced back and forth between Anhaica and Ivitachuco). When de Soto encountered the Apalachee, Capafi appears to have been their nico, or principal chief, and Apalachee warriors fought hard to successful rescue him from captivity. But by 1608, when a Spanish priest helped negotiate a peace treaty between the Apalachee and the Timucua, Fray Prieto emphasizes the role of the holatas, or subsidiary chiefs, of which there were 40.

The micco of Coosa appears to have been the leader for most of the southern Appalachians, from North Carolina into east-central Alabama when de Soto came through. By the 1700s, however, Coosa was just one of the four founding members of the Creek Confederacy and its micco one of the delegates serving on the national council.

Contact era Dhegiha culture, the presumed descendants of the Middle Mississippians such as Cahokia, also had a national council structure. Admittedly, for the Osage at least, they did have two gahige serving as figureheads of the two moieties. But it was ultimately the council of Nonhonzhinga (elders) that had the real power.

For whatever reason, there appears to be a trend beginning before European contact and continuing afterwards of rejecting centralized powers in the Eastern Woodlands. With Euroamericans and Europeans following suit and adopting more egalitarian political structures later, we might say these Native communities were ahead of the political curve on this point.

The fact that they didn't develop similar technology with the Europeans is explained by the fact that they strayed away from centralised, agricultural states

I just wanted to add that, when it comes to metallurgy in particular, there are issues of geology and physics to consider. The Old Copper Complex in Wisconsin and Michigan were making metal tools and weapons around 6000 years ago. The purity of the copper deposits there meant they could work it without needing to smelt it first, and generally speaking, it appears societies need to be smelting copper (which requires lower temperatures) before they can begin smelting iron. Otherwise, you don't have the smelting infrastructure built up to bother achieving the temperatures necessary to smelt iron. Of course, West African ironworkers may have skipped copper-smelting and jumped straight to iron so it's not a guaranteed step-by-step process, just an unlikely one.

Of course, the Old Copper Complex's metal tools and weapons were somewhat flawed for the same reason they could be produced in the first place. By itself copper is too malleable to hold an edge for long. So once long distance trade really kicks off in the late archaic (circa 3000 years ago), Old Copper switches back to stone tools and weapons and shifts copper production to ornamental and ceremonial designs.

So I wouldn't ask: why didn't North American societies develop similarly to European ones. I would rather take a step back and wonder: What prevented the agricultural, centralised-state society from spreading in the region of North America?

These are the same question--you're still assuming an agricultural, centralized-state society as the default, which America was somehow "prevented" from achieving.

What you're interested in, I think, is the political history/geopolitics of how Native American societies evolved over time and in contact with each other.

I'd suggest you submit a new question focused on that specifically. The question about Eurasia would better be separate--you're more likely to get answers from experts in either field that way.