I suspect that some parts are missing. Cyclo made 2 and 3 speed rear sprocket clusters that could be installed on Sturmey Archer and Sachs 3 speed hubs. They also might well fit old Sachs coaster brake single speed hubs. They also made a plunger type derailleur and mounting bracket to work with their sprocket clusters.

Sachs themselves did make derailleurs to work with some multi sprocket coaster brake hubs. They included an upper chain guide to retain the chain when reverse pedalling to operate the coaster brake. IIRC that setup is shown in the third edition of Frank Berto's book "The Dancing Chain". BTW it is an excellent reference on many oddball bicycle gear change systems that have been used.

From what I have read German city riders demanded that their bikes have a coaster brake hub and Sachs built derailleur and coaster brake setups to meet the demand.