Central Ohio family here! I've been in the area for close to 20 years when I moved here for graduate school and ended up getting married and staying. It's a wonderful place to raise a family and very affordable, although the housing market is crazy right now! Lots of bidding wars, at least in my particular suburb.
We're embarking on our 3rd year homeschooling this year and have found there are lots and lots of opportunities for homeschoolers all over the city. We do not participate in a co-op, but there are lots to choose from, though I have found it difficult to find many classically-minded groups. This year a small group of us will be meeting together once a week to take our 3rd-4th graders through MP's Latina Christiana, which in addition to homeschool gym (Coach Q runs an awesome program all over the city, definitely look into it if you move here!) and some other activities gives us plenty of learning and socializing with other families. I don't have much experience with older homeschoolers since my oldest is only going to be in 3rd grade, so I can't give much guidance on Columbus State, but I have a close friend who teaches full-time there and my husband is an adjunct there and it seems like a good option from my limited experience.
In regards to notification laws, another option to the portfolio assessment or test scores is to form an 08 school, which is a non-chartered, non-tax supported school. It's different paperwork you need to send to the state DOE and your local school district, but you don't need to send in any form of assessment for your kids, just their names and addresses. If you aren't legally inclined, I'd suggest joining HSLDA for a year because they have guidance and form letters that follow the 08 school statute requirements. It's not hard, you just want to make sure you do it properly.
And I have to put in a plug for the children's hospital since I saw Cincy was mentioned lol - I lived at the one in Columbus a few years ago for 6 months while my baby boy was inpatient for chemo and a bone marrow transplant for high-risk leukemia and that place is amazing. They literally saved his life multiple times during treatment and we continue to go there frequently for all of his follow-up care. ☺️
Feel free to ask any more questions you may have about the area!!