footway tag is usually used for ways inside civilized areas like cities.
path tag is more related to hiking and backpacking activities.

regarding the "S" this is how it is represented on USGS Topographic maps.

it is actually continuous now, but not directly.
USGS Topographic maps and satellite imagery shows that we can re-access Batona Trail via Lower Forge Road and Goodwater Road.

and this is what is actually on the ground.