The .375 Winchester has been on the ODNR list of approved straight walled cartridges prior to the 2017 change that permits any straight walled cartridge of .357 caliber minimum to .50 maximum & should work just fine.
The only issue with the .375 Winchester is the limited availability of ammunition & components & rifles chambered for it. I would love to find a reasonably priced Savage 99 in .375 Winchester for Ohio use & addition to my Ruger #1 .405 Winchester.
.375 Win. Yum.
The Savage 99 would be great.
However, this all rear-view-mirror stuff. There are Win 94's out there and a few Marlins, but no current production. The other problem I see is the ammo. Unless you're loading your own and casting your own, the ammo that's out there is more fit for a moose than a whitetail. Most of the factory bullets are for 375 H&H and they're meant for big stuff and dangerous game.
That's the other thing I see with these Ohio regs: If you transpose it onto the list of what's really out there, the choices of available ammo are mostly all shoulder busters. I'm no wimp, but 3 rounds of full-house 45-70 factory ammo usually is the highlight of my afternoon. Most of the others leave a serous mark as well.
375 WIN is better in that regard. In fact, recoil-wise, it's getting right in folks' comfort zone. I'm hunting 10 miles from the Ohio River on the south side. We get to fling everything we want at the little critters. Gimme a nice 30-06 any old day. 375 WIN has about the same punch.