The simplest explanation for Ohio's list of "approved" straight-walled cartridges is that it's incomplete.

The scenario here will be much like what happened with Ohio's CCW law, which each year since it passed (& thanks to the ever-vigilant Buckeye Firearms Assoc) got "tweaked" with amendments that added this or that clarification or improvement and deleted this or that bone-headed provision which never should have been in it.

Similarly, this ODNR regulation will no doubt see future amendments adding more cartridges to the list.
For example, if the .357mag & .38 Spec Ed got approved, why not the more powerful and larger bore 10mm AUTO?
If the response is that no "lever" longer guns are so chambered, that's just a matter of time or, as a custom proposition, money.

For what it's worth, and like the dual usage capability of the .38Spec./.357Mag or 44 Spec./44Mag in wheelgun form, you can also shoot the .40S&W out of a 10mm S&W 610. You could do that with a lever gun too, as far as I know. In other words, with some of these straight-walled cartridges which have smaller siblings, you get the benefit of being able to shoot a second cartridge in the same gun.

All that said, I do my deer hunting with a rifle, most of which has occurred in upper Michigan and the U.P.
Among the approved cartridges on the current list, my choice is going to be the 45-70 simply because I know where a hardly-used 22" Ruger #1 is available for a reasonable price and, after mounting a 1-4x Leupy and zeroing it for 100yds (the farthest I'll be shooting off a tree-stand overlooking a buddy's cornfield), that set-up will be good to go for as many deer as Ohio will sell me tags.