I know that it comes from "chef," the French word for a head person (like in a kitchen), and that many of the early accounts of aboriginal Americans were written by the French. Still, why did they choose that word and not another, and why did speakers of other languages pick up the French word (if indeed that's how it happened)?
Chief was originally used in English meaning "the principal or most important part" (circa 1300).
In the 1500s it grew into a noun meaning "tribal leader", in relation to the Gaelic clan systems of Scotland and Ireland. It was later used in relation to American tribes as a natural extension of this definition.
This answers your question, but I think a more interesting question is:
When and why did "chief," as opposed to any other high-ranking title, become the standard English noun applied to Gaelic clan leaders?
Also, why is the president called commander in chief? Do other English speaking nations use the word or did the USA take it from native American terminology?
The use and spread ofcacique in parts of the America that would later fall under Spanish dominion is instructive. Cacique is a "spaniziation" of Taino term for a leader. The earliest Spanish arriving in the Caribbean dealt primarily with groups that used the term, and so it became widely adopted for any elite American individual.
By the time of the Cordoba expedition to the Yucatan in 1517, the term had gained ground and was generally used for any indigenous person of authority. So we don't have account of the the Cordoba, or subsequent Grijalva expedition, meeting Ahaus/Ajaws of the various polities they stopped at, instead we have cacique.
Similarly, when Cortes arrived on the Gulf coast, he called the leaders there cacique, and not whatever the Totonac term for leader was (I confess to not knowing). When he moved inland to the areas where a Tlatoani had elite Teuctli who had more subordinate Pipiltin, this nuance was likewise not kept. A tlatoani might become a "king," but the rest became cacique in the accounts. This blurring of distinction would last into the colonial era, as individuals who might have formerly in any of the other classes (noble or common) began to adopt the term cacizago as a descriptor for themselves.
If you think about the the French and English colonialists arrived further up the coast of North America, they had even more linguistic confusion to deal with. Mesoamerica had a dominant trading language (Nahuatl) and some over-arching language groups to the extent that Cortes was able to get by with a Spaniard who learned Mayan translating with an American woman who was fluent in both that Mayan dialect and Nahuatl.
The colonialists up north, however, had a much broader range of languages to deal with. The problem was compounded by the mixing and reshuffling of groups at the time due to epidemics and re-absorption of groups. So, in a certain sense, adopting a universal "chief" to denote the elite of any American group made some sense by creating a linguistic category for "elite (other)" that could be broadly applied without learned the term or pronunciation of the local term.