I imagine it's because, for it to be the Missouri River, it's making an awfully sharp turn at the point where it meets the not-Mississippi River, and anyone looking at a map would ask "Why isn't that part of the not-Mississippi River? That would be so much more natural."
Besides which, Missouri has several pro sports teams, the Arch, the Ozarks, the Missouri River . . . it even has
me. Mississippi has the Mississippi River aaaaand . . . um, well.
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“There must have been a point in early human history when it was actually advantageous to, when confronted with a difficult task, drop it altogether and go do something more fun, because I do that way too often for it to be anything but instinct.” -- Isto Combs