Off the top of my head, I can’t think of a word in English that ends with “is” while being singular, only plurals and uncountables come to mind, so I can’t really follow the examples of other words. What makes it even weirder, I’m not sure how to pronounce Illinoises… Would it be as written, or as if an Illinois was pronounced by someone who has never encountered it before? Illinoi are also meh, since now plural looks as a singular and the other way round.
It did not start out as an English word. It came from the Miami-Illinos language to Algonquin and/or Ojibwe to French then to English. So applying English language rules to a non-English word will be difficult.
But as AbouBenAdhem suggests ITT, treating it like the French word “Quebecquois” makes a lot of sense.
Info about the etymology:
https://www.etymonline.com/word/Illinois
https://theculturetrip.com/north-america/usa/illinois/articles/this-is-how-illinois-got-its-name
https://en.m.wikipedia.org/wiki/Illinois
https://en.m.wiktionary.org/wiki/Illinois