No. (at least not the points I think you're talking about)
FS's get Fellowship Experience Points (XP to spend on leveling up Perks) if they finish the FA in 1st-100th place. On browser you can click the FS banner in the top left of the FA map, then hover over the last icon in the rows and see the number of XP points awarded for each position.
The Waypoints on the maps for Stages 1, 2, and 3 give FA points. Those are what total to the FA score you see next to your FS name on that same window.