The history of Arsenal Football Club shows that the self-sustaining model has not always been successful for the club. There were several years of financial problems which led to Sir Henry Norris' involvement and the transformation began with the formation of a new Woolwich Arsenal Football and Athletic Comany as reported below.
The story is covered in more depth in the soon to be published book, Woolwich Arsenal F.C.: 1893-1915 The Club That Changed Football by Tony Attwood, Andy Kelly and Mark Andrews, which can be purchased here.
I am indebted to Andy Kelly for providing the scan of the Kentish Independent article at the bottom of the page, which is a simply fascinating read. Modern football politics has nothing on this!