The line shapefile is obtained from the Dissolve toolbox of all routes, so there is only one polyline in this shapefile. The point shapefile is already there. The Split Line At Point should allow me to get another shapefile which will be split according to the points. I notice that some of the lines are split, but some of them are not. The common feature for those lines don't get split is there is no turning line at that point, as shown below. This means the line is split perfectly when the point is the intersection, but for those points are just in the middle of a line, the split function will not work. Also, the points used to split the line are obtained from this line shapefile, so they are perfectly located along the line. This is a new question, which is not the same as the similar questions asked before since the problematic points are all the middle points along the line without intersecting with other lines. Any comments on this issue?
The following picture shows the problematic points where the line can't be split. I am using ArcGIS Desktop Advanced License 10.3.1.
What happened when you tried "to make a copy of your data and run the Integrate tool first to ensure the lines and points are fully snapped before trying the Split Line at Points tool"? I think you may find that your points are only of near perfect alignment - Integrate should convert that to perfect alignment and enable the split.