Calculate a distance from end point of street line to nearest point. There are a couple of ways to do this.
Simplest is to do an Aggregation of Street and points with 'accumulte nearest' with output
min_dist - MIN(DISTANCE(Input.Geometry, Detail.Geometry, ProjectedMeas, Meter))
Then 3 functional Attributes using this result
Remainder - Input.min_dist
Endofline - ENDPOINT(Input.Geometry)
End_of_LIine - MOVEBYDISTDIR(Input.Endofline, Input.Remainder, ORIENTATION(Input.Endofline) +180, ProjectedMeas)
New_Line - CREATEPOLYLINE(STARTPOINT(Input.Geometry), Input.End_of_Line)
This should create a query of the new lines, you can then save.