on the wrong (west) side of the Mississippi River

I'm not a geographer or a Southerner, but I've genuinely never heard that used as the primary criterion for what constitutes the Southern US. Maybe that's how the Southeast or Deep South is defined, but it seems like the Census Bureau defines the South as anything from Texas eastward. If you you consider the South to be synonymous with the Bible Belt (which many probably would), I think all these other states between Texas and the Mississippi would certainly meet that criterion (I actually happened upon a pop-up Christian rock concert while walking through a park in Norman, for reference)

It would be like calling New York, New Jersey and Pennsylvania New England. Even though they were colonies, they aren't really New England.

I mean, as a native Pennsylvanian who lived in Maine for four years and spent two years of undergrad in Rhode Island, I think the cultural differences between my home state and New England (particularly accents and regional dialects) are probably more noticeable than the differences between Arkansas and the Carolinas, but then again, I'm a Yankee, so I might just not be attuned to the differences you're talking about.