One idea you can try - it is not snapping but creating points at the nearest locations on the lines:
- Run the Near tool on your points, with the "Location" option. This will add the NEAR_X and NEAR_Y fields to the point input.
- Then run the Make XY Event Layer tool, using the point feature class as the XY Table and the NEAR_X and NEAR_Y fields as the X and Y Fields. This should give you a point layer with all the points on the lines that are nearest to the input points.
- You can run Copy Features to save this layer to a feature class; all the input point attributes are carried over.
Hope the spatial associations of your points with lines are clear. If any point is in the same shortest distance to more than one line, then you might get a point on the wrong line.
Thanks.