I think it’s not a great illustration; the map looks like mercator so the shortest path would be curved and closer to the green path anyway. the red line is not “as the crow flies”. It would be more fair to show deviation from the actual (great circle) shortest path. Also agree the grid seems coarse.
The red line is the great circle path according to the map legend.
Are we looking at the same map? On the one I see, showing western and central Europe, the difference between a great circle and a loxodrome (straight line on a Mercator map) for an intra-European route would be barely visible. And the map is clearly not Mercator — France looks visibly "smashed". Looks like a simple x=longitude, y=latitude projection.