If finances are an issue it would be a no brainer to choose OSU over Michigan.
My Ohio raised daughter did not have a good OSU visit. Felt her major was too grad focused and not enough undergrad research opportunities. She also found it too big. We also had other things going against us - we were hit up by a group of panhandlers outside the student union (not normal from what we’ve heard) and our tour guide was just not good - just didn’t seem to like the university very much and wasn’t a motivated student. She also felt it was “too close” to home and knew too many people going there. And, she wanted a more suburban/rural setting. D was on campus multiple times throughout HS and just was underwhelmed each time.
Our Michigan visit was one of the best. The engineering tour hands down was the best of any school we saw (we made 15 school visits). Our tour guide was in the same major (you could pick which guide you went with at Michigan), was doing research that was interesting to my daughter, and they did a very thorough engineering facilities and lab tour. There were some negatives for her too but it was her #2 school choice. Had she not gotten into her top choice, she would have stayed on the WL and done everything she could have to come off.
Do I think job and grad school options will be limited by going to OSU honors? No I don’t.
Note: some students in OH wouldn’t apply to Michigan if their lives depended on it. Too big of a rivalry.