Across Turtle Island, it is extremely common for power to be held in separate spheres. Many Algonquin peoples would have war chiefs that exclusively had authority in warfare while regular “governing” affairs and even diplomacy would be done by elders. Aztecan peoples had elaborate specifications and restraints to an individual’s power, with the Mexica having two rulers, the cihuacoatl for civil affairs and the tlatoani for war alone (the latter is still used as a synonym for the President of Mexico). Siouan peoples emphasized the authority of grandmothers; this was widely brought to the radical consciousness during the Standing Rock demonstration camps.

The Haudenosaunee are a matrilocal and matrilineal culture. Councils of matriarchs are the ones that make most of the decisions, including land management and even the recall of war chiefs. At some levels historically it was exclusively matriarchs making these decisions, and at other levels, especially more recently, wider assemblies would be more common.

Europeans coming to observe these cultures were not able to understand the nuances of these, and imposed their ethnocentric understandings of “king” and later “leader” onto their observations of indigenous societies. So while the colonists would have engaged with a war chief, it would have typically been a matriarch or group of elders whose will was represented by that war chief.

Not giving an opinion either way but just going to point out that, in those quotes, they’re referring to direct democracy (the reference to Athens, in particular, is intentional), not any kind of democracy (e.g. representative democracy).