Nah. The Blue Jackets are Columbus, not Ohio.
Columbus is a younger city largely made up of transplants from the other C's. I myself am mid-30s and from NE Ohio, so my other sports rooting interests are firmly Cleveland except football because I grew up in Canton and hate it. Columbus is growing at a time when the other two are shrinking, and it's good that Columbus has its own team in one of the major four sports (not to mention one of the most historic MLS franchises, even if ithat league is a distant fifth in popularity in the state). Even then, fandom for pro teams here is more heavily weighted toward Cleveland sports, especially among people my age. Cleveland and Columbus just seem to have more in common with each other than Cincinnati and Columbus, and not just because of the Guards/Clippers and Monsters/Jackets connections.
Ohio is definitely not homogeneous. There's like 12 million people in this state but we fall into four very distinct quadrants and there isn't much overlap between them except Columbus where they converge (Appalachian SE, Rust Belt NE Ohio, Faux-Southern SW Ohio, and NW farmland). NE and SW have obvious allegiances, SE tends toward Pittsburgh sports because that's who was on TV/radio there, and everyone I know from NW Ohio is a Browns/Red Wings/Yankees fan for some reason. Then you have micro demographic trends, like quite a lot of NE Ohio Millennials are Steelers fans because they're nearby, a) the Browns actually didn't even exist for a big chunk of our childhoods and b) when they have existed, they've sucked. A lot of people from SE Ohio switched to the Bengals because of Joey Burrow. OSU fandom stretches across the state, although mostly concentrated the closer you are to the capital, and even then OSU fans tend not to be professional fans and vice versa (there's also a lot of bad blood between OSU and the Jackets).
All that to say, there's no point in re-naming the Jackets to the "Ohio Jackets." They're not Ohio's team and never will be, and that's okay.