Are you able to use the start or the endpoint of the line as start points? does direction matter?
If there is any logic that can apply to the points' distance with tolerance then it might able to do it through python.
For instance, I will get all the points along the line or interests with the line. Then calculate point distance from one direction to compare every point's distance to the start point of the line and to the endpoint of the line...
loop through all the points then you will get the list of the points based on the comparison from one direction.
~Vicky Wang~