I'm the OP (local to the area). Not much to post update-wise, other than the family organized a search party in the battlefield this morning. It doesn't appear that they've found anything yet. Weather-wise, it rained pretty much all day Thursday (yesterday), so I'm sure some of those trails are a muddy mess.
My husband and I drove around for hours, looking for skid marks or any tire tracks where a vehicle may have left the road, but found nothing. Others have done the same. Nothing.
There still doesn't seem to be any information about the identity of the man who was helping him fix the vehicle. Nobody has come forward. If the vehicle was seen at the visitor's center, where lots of older guns and historical artifacts are on display, it would seem that there would be security cameras. (??)
Here's a link to the trail map, if anyone is interested. The more difficult trails are on the northwest side (you can see the terrain on the map), but even with my bad knee, I've been able to hike most of them. Also, the horse trails are generally wider and easier than the foot-only trails.
https://www.nps.gov/chch/planyourvisit/upload/Chickamauga-Battlefield-Trail-Map.pdf
Here are a few random photos I've taken of the trails in the last couple of years. These are all easier/wider trails, but you can see the surroundings and how muddy some of the trails can be after it rains. Some trails are also rocky and root-y. A few are steep. The woods get VERY dense in some places. As far as potentially dangerous wildlife, there are a crazy number of deer, but they run if they feel you're too close. Coyotes have been seen/heard, but I've never heard of park coyotes being aggressive toward people. As far as I know, no bears have ever been seen in the battlefield. There are small streams that meander through the park, and, as mentioned, Chickamauga Creek runs along with southern border.