Lets go back to what you defiantly know.
John F was born 1/8/1911 in Sydney.
Father Joseph was described as a Grinder Engineering Works.
Not at Lithgow, the factory hadn't opened at that time.
So how many engineering works were they in Sydney at that time?
He appears to have been a workman, not super skilled so I don't think he was involved in setting up the factory, so I think you can discount that story.
The Birmingham marriage I think you can be sure that didn't take place, at that time no person would have dreamed that in 100 plus years in the future their descendants would be able to check out easily any story they told.
I guess Joseph or whatever his original name was had been married in the UK and had taken up with May and they left for Australia possibly changing names en route.
Have you tried DNA? It could possibly throw up some clues.