Ohio law treats this as a live question, and the answer turns on how the two documents are drafted, which is exactly why the strategy guide stresses keeping the option and the lease as separate contracts.
In most Ohio cases, if the lease and the option are written as independent agreements, terminating the lease does not automatically extinguish the option. The option is a separate contractual right, and the tenant-buyer may still hold it even while in default on rent. That is the mess you are already picturing. To kill the option, you would typically need a separate legal step, most often a lawsuit to quiet title or a declaratory judgment removing the option from the record.
If, on the other hand, the two documents contain explicit language tying option survival to the lease being in good standing, a court is more likely to treat default as forfeiting the option as well. The drafting is doing the work here, not some default rule.
I want to be direct about my limits: the specific outcome in your situation depends on Ohio contract law and how a particular court reads your documents, so you need a real estate attorney licensed in Ohio before you sign anything on a sandwich deal. That is not a detour, it is the actual next step.
The thing worth pricing for before you get to that call: if you are the middle party in a sandwich lease option, you are exposed on both ends. Your option to buy from the seller and your lease to the tenant-buyer run on different timelines and different terms. A tenant-buyer default can leave you holding obligations to the seller with no income covering them. That asymmetry is worth modeling before you structure the deal.
What do the option terms in the Columbus deals look like so far, specifically how long the option period runs and whether there is any good-standing language tied to it?