These empires weren't neolithic, and it's very debatable whether there was a negative technological disparity compared to Europe (which you seem to be using as your point of reference), outside of areas like siege weaponry, shipbuilding and armour smithing . Could you rephrase your question?
While population density is limited by technology, i.e., farming practices, modern tools are able to make certain tasks take much less time (i.e., metal axes and machetes were prized among natives because steel was much more resilient than stone/bronze/copper, in fact, stone axes were still used for cutting because while you could make bronze sharper, it wouldn't last that long).
However, if you look at certain civilization tasks, while you might consider them as "technologies", they don't necessarily depend on ability to process metals and quality of your tools.
For example:
Good irrigation and natural fertilizers can make piece of land highly productive, while less intensive techniques can make land production barely sustainable. But if you have good organisation, you can transform land, make irrigation canals, bring water with aquaducts or organize enough men to clear jungle (clearing jungle to make a field is very laborious). Just look how intensive land management could turn some desert areas into breadbaskets, supplying wheat into whole Mediterranean area, while after organisation was disrupted, they turned back to arid area with low yields.
Sanitation is another such example. You might remember very bad sanitation in most European cities, while Knossos palace already had running water in ancient times. In fact, in most places even in Europe, "night soil" was collected and used as fertilization, the same was done by Aztecs. If you dedicate enough infrastructure and manpower to manage this, something that you don't need iron tools for, you can easily make place much cleaner, decreasing risk of disease, while using what you collected as fertilizer, even increasing yields. In fact, when Prague wanted to upgrade its canalization system, a lot of farmers protested because they were afraid that they will lose access to a cheap fertilizer.
Finally, to take advantage of large population, you don't really need advanced metal tools, you just need developed bureaucracy to manage it, write down who lives where and who gave how much in taxes.
As I hope I demonstrated, to have developed empire with high population density, you don't really require metal tools. And while there would be limits where gain would be not worth of additional tool because efficiency dropped too low, which could be modified by having a more efficient tool, population density could reach relatively high levels even without these tools if you make required organisational steps and structure.
Aztecs had developed irrigation system, with dams and elevated banks of several lakes around Tenochtitlan, they collected feces for irrigation and urine for industry and had developed organisation with high control of citizens, prescribed who will support whom and state guaranteed food and goods supplies for time of need.