Garmin shows Courthouse Gap Rd as getting close to Rock Mountain, but Google Maps says this road is much shorter. I found a “Private Road” sign on it at N34.91567 W83.35029. I don’t think you can get to the Courthouse Gap Trailhead at N34.91567 W83.35029 unless you find the owner of the private part of the road.
Alternate trailhead at N34.90724 W83.33531 at the end of Finney Creek Rd has a longer bushwhack. After turning off the paved Warwoman Rd, the road turns to twin-track, so drivability in 2WD is doubtful.