The Australians had a long history of acquiring and using whatever arms they thought best for their needs and used just as many American firearms as British ones right up until WW2. Colt and Smith & Wesson revolvers were just as common as Webley's and other British made firearms in use among the police and Militias and with the public.
Australia never intended on acquiring the Ross or used that as a excuse to build their arms factory. The British wanted all their colonies to be self sufficient and equipped with the same arms and even set up arms factories in what is now India and Pakistan. Australia became independent of England in 1900 with the Commonwealth of Australia Constitution Act of 1900 which became law on the 9th of July 1900, way ahead their building the Lithgow factory or of Canada which waited until 1982 to do the same.
Australia was long a independent commonwealth country when Lithgow was built and that is why they paid for it themselves.
The Australians choose the Pratt & Witney machinery because it was proven to be the best mass produced arms making machinery available at the time and the price was less than the British Machinery which was slower and required far more workers and hand fitting to produce the same amount of completed firearms.
The finished Lithgow SMLE's were regarded as first rate and as good if not better than any out of England. Their steel quality was also first rate as Australia then as now has some of the purest ore in the world. Most of the high quality steel coming out of Japan and China now is made from Australian ore.
The Lee rifle is also a American design by a Scot named James Paris Lee who settled in America and became as citizen there. His rifles were adopted by both China and the USA in limited numbers before the British. They were made by both Winchester and Remington before England adopted it and had Enfield modify it to meet their needs with longer lasting Enfield rifling, hence the name Lee Enfield.
Here are some early American made Lee rifles -