It comes down to how the ELL extension was funded. The only way money could be secured for it was to get a private company in to run the line after it reopened and become part of the National Rail network. (This is why you see the double arrow sign on Journey Planner)

London Overground while part of TfL is operated by London Overground Rail Operations Limited (LOROL) under a franchise agreement. This is a similar arrangement to how the DLR is operated, which is run by Serco.

The company is a 50:50 joint venture between the Hong Kong metro rail operator MTR Corporation and the German rail operator DB Regio (a subsidiary of Deutsche Bahn AG).

Unlike Southern, the Overground franchise is decided by Transport for London and not the DFT.