According to The Cornell Lab of Ornithology, American Robins are indeed found in wild places like woodlands, forests, and mountains. Because worms hibernate, the winter robins feed primarily on berries, found on trees and shrubs in the woods.
The Ohio DNR Division of Wildlife lists robins as native to Ohio. Some migrators travel through but there is a population that overwinters, in both urban areas and woodlands. They're shy and not easy to spot, which is probably why you haven't seen them, but it's worth looking.
It sounds like you're in a great spot! In fact, the Columbus and Franklin County Metro Parks Agency oversees 17 natural parks with 200 miles of trails and over 27,000 acres of land in Central Ohio, one of which is Inniswood park. The agency is headquartered in Westerville, so hopefully that's close enough by for you to check out the area. There are a number of places for bird-watching.
A group called Trek Ohio keeps records of where the robins overwinter in central Ohio. Sightings have risen into the thousands in Inniswood. They have an interesting article Ohio's Overwintering Robins. Here's a picture from that article. It's a robin foraging berries in the woods.
This one's from the Ohio DNR site referenced above:
Get some binoculars, poke around, and let us know what you find!