I've made a python script for that. Basicaly, we found the nearest point from all the points on our line feature. We group the created point by pair, then we create a line with those pairs. The last line add the created line to our initial line feature.
fcpoint = r'mypoint'
fcline = r'myline'
database = r'mydatabase'
# Set the workspace
arcpy.env.workspace = database
arcpy.env.overwriteOutput = True
# find features only within search radius (no limit here)
search_radius = ""
# find location nearest features
location = "LOCATION"
# avoid getting angle of neares features
angle = "NO_ANGLE"
# execute the function
arcpy.Near_analysis(fcpoint,fcline, search_radius, location, angle)
layer = arcpy.MakeXYEventLayer_management(fcpoint,'NEAR_X','NEAR_Y','point_on_line_temp','2056') #don't forget to precise your own SRID
arcpy.FeatureClassToFeatureClass_conversion(layer, database, 'point_on_line')
arcpy.Append_management (fcpoint, 'point_on_line', 'NO_TEST')
arcpy.PointsToLine_management('point_on_line', 'line_from_point', "UNIQUE_ID_OF_MY_POINT", "", "NO_CLOSE") # Don't forget to give the unique ID of your point feature.
arcpy.AddField_management('line_from_point', 'CONNECTOR', 'FLOAT')
arcpy.AddField_management(fcline, 'CONNECTOR', 'FLOAT')
with arcpy.da.UpdateCursor('line_from_point', 'CONNECTOR') as cursor:
for row in cursor:
row = 1
# Remove this line if you want the connected line as separated feature, this line merge the created line with fcline