There are floating point precision errors when finding a point on a line. Use the distance with an appropriate threshold instead.
from shapely.geometry import Point, LineString
line = LineString([(-9765787.9981184918, 5488940.9749489054), (-9748582.8016368076, 5488402.1275707092)])
point = Point(-9763788.9782693591, 5488878.3678984242)
line.within(point) # False
line.distance(point) # 7.765244949417793e-11
line.distance(point) < 1e-8 # True