The "Hamburg Steak" became popular in the late 19th C. I found mention of a recipe in an 1899 Australian cookbook. (The earliest US recipe I found is 1844)
It's the "hamburger sandwich" that Vinny is talking about. There are all sorts of claims about its origin. According to Wikipedia, "all claims made by the potential inventors of the hamburger occurred between 1885 and 1904." Early ones were just sandwiches, but ketchup became the condiment of choice by the early 20th C. and the full lot of mustard, pickles tomato, lettuce etc. was in place by WWII.
Apparently the American-style hamburger really took off in Oz in the 1970s when McDonalds & other large chains expanded into that country.
What I can't get a good handle on is when beets became popular. I found mention that Americans stationed in Australia during WWII were surprised to get hamburgers with beets. One reference said that the Aussie style began in the 1930s.
(In 1969, Pan Am Airways produced a "complete guide to travel in the United States." They described hamburgers:Grilled or fried ground beef patty, nearly always served on an unbuttered bun. It needs catsup or a slice of onion before it has much personality.)