Ohio's courses have had the option to remain open throughout the crisis, aside from about a 36 hour period a few weekends ago. And even then, most private clubs that I'm aware of were able to keep their courses open.
Our state's leaders continue to advocate for socially distanced outdoor recreation, and golf has been a good outlet.
Walking is definitely more prevalent and most courses seem to be actively endorsing it. But I'm not sure it's going to spur a walking golf renaissance among the general public. The most walking-friendly public course routings in town are munis, and they're all closed (unless something changed recently). Meanwhile, I saw a residential course across town encouraging walking on their website a week or so ago. I just measured the distance from their 9th green to the 10th tee at 2200 feet on Google Maps, and that's in a straight line. The actual route is probably closer to half a mile.
It's been interesting to watch round times go down as everyone playing my home course started walking it, but that ain't gonna happen everywhere.