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There are multiple toolboxes in ArcGIS Suite that deal with geographic transformation and correction, such as Spatial Adjustment, Rubbersheeting, and Snap. I'm not sure how to best use a tool to correct my line to a series of points as shown in image #3:

1) is what I currently have;

2) is what I get when I use the Snap tool; and

3) is what I want.

The points are considered accurate geographic "anchors" for the lines, but don't otherwise share any field attributes with the lines. I have access to a list of all of the point coordinates.

If I snap the line without vertices with the Snap tool, the tool doesn't work at all, even with a high tolerance set (higher than the distance between the line and the point).

I'd like to stick to ArcGIS Suite, and to use something that has a corresponding arcpy tool. I realize that I can move the anchors/delete the middle anchor (vertex) to get the shape in Figure #3, but I plan on automating this process because I have hundreds of lines and thousands of points.

Append end points of your lines to snap points and run near tool on appended set of points. Add field type "Double" to their table:

Rename original lines in table of content to "original" and use field calculator:

g = arcpy.Geometry()
geometryList=arcpy.CopyFeatures_management("original",g)
def getChainage(lineFID,point):
 line=geometryList[lineFID]
 return line.measureOnLine(point.firstPoint)
#------------
getChainage( !NEAR_FID!, !Shape! )

To populate new field in points table:

Points to line tool will do the rest:

arcpy.PointsToLine_management("points", "../SNAPPED.shp", "NEAR_FID", sort_Field="CHAINAGE")

Solution valid for shapefiles, it is a bit more complicated for other storage options. Note that you can use linear referencing to compute distances of points along original lines if you'd like to avoid field calculator used here.

I would approach this by reconstructing the line manually. Use a cursor to extract the start and end points from the line, sort the list of point coordinates by proximity to the start of the line, and reconstruct the new line geometry.

# assumes one line in in_line
in_line = r'scratch.gdbsample_line'
in_points = r'scratch.gdbsample_points' 

# get line start and end points
with arcpy.da.SearchCursor(in_line, ["SHAPE@"]) as cur: 
 for row in cur: 
 geom = row[0]
 point_first = geom.firstPoint
 coord_first = [point_first.X, point_first.Y]
 point_last = geom.lastPoint
 coord_last = [point_last.X, point_last.Y]

# get point coords as list
with arcpy.da.SearchCursor(in_points, ["SHAPE@"]) as cur: 
 points = [[row[0].centroid.X, row[0].centroid.Y] for row in cur]

# define a function used to sort points based on proximity to point_first
def coord_dif(coord_x):
 x_dif = abs(coord_first[0] - coord_x[0])
 y_dif = abs(coord_first[1] - coord_x[1])
 return (x_dif + y_dif) 

# sort points
points = sorted(points, key=coord_dif)
all_points = [coord_first] + points + [coord_last]

# construct geometry
line_geom = arcpy.Polyline(
 arcpy.Array(
 [arcpy.Point(pt[0], pt[1]) for pt in all_points]
 )
)

# update with new line goemetry
with arcpy.da.UpdateCursor(in_line, ["SHAPE@"]) as cur: 
 for row in cur: 
 cur.updateRow([line_geom])