Just about every other society developed some kind of written recording/writing styles. How come the Native Americans didn't?
They did. Wiigwassaback was developed by the Algonquin, and they made codexes from birch bark. If they were not the first to use birch bark as a medium, there may have been prior writing systems, but they are lost to us as soft tree bark in a humid climate is an impermanent medium, unlike baked clay or stone, or papyrus in the desert.
There are a whole lot of societies who didn't develop writing. I don't have enough knowledge to give an extensive list, but many sub-Saharan societies, most Polynesian and Melanesian societies, even the Inca didn't have a "true" written language (despite their proximity to the Maya), but used mnemonics such as tally marks in the form of knotted ropes.
The premise of this question is deeply flawed.
There's always room for discussion, but perhaps this previous topic will answer your inquiry.