Now... who’s gonna believe there is a Kansas City, Missouri AND a Kansas City, Kansas ? They would probably think it a typological error.
When I saw the location of the school being described as south of Kansas City, I found it to be confusing. I think of Shawnee and Mission, KS, as being just west of Kansas City, MO, across State Line Road. When I see a city described as south of Kansas City, Grandview and Raymore past the Triangle come to mind. I believe a clearer description of the location of the school would be either that it is west of Kansas City, MO, or south of Kansas City, KS. But as you suggest, most people don’t know, or even care, whether there is a Kansas City, KS, it being in flyover country and all.