This might blow your mind:
Forest Rd, which runs through much of the St George district, when it was first built was called Illawarra Rd. So from about the mid-18th century onwards, the area was known as the "Illawarra Suburbs". This still lives on in the Illawarra Suburbs Lawn Tennis Association - based in Rockdale! So, in the early decades, the St George rugby league team were often referred to as the Illawarra men, or Illawarraites
I think you mean the mid 19th century. Mid 18th century was pre-invasion.
Back in the 1800s it was a timber getting area. And you're quite right about it being referred to as Illawarra. It seems that everything south of the Cooks River was called Illawarra. There's strong evidence of this in street names in Arncliffe (eg Wollongong Rd) and in Marrickvlle there's Illawarra Road which leads down to the river.
The Parish of St George was becoming densely populated in the early 20th century and the district was simply known as St George, and that naturally became the name of the football team. I've only ever been able to find one written reference to the team nickname 'the Illawarraites' - if it's true then the name didn't stick and the 'Saints' nickname simply rolled off the tongue better.
But there's enough there with the history of the area to support the notion that the new Rugby League club could have been called Illawarra. And if that happened, everything would have been very different indeed.
One thing we do know is the St George club has had strong ties with Rugby League in the Illawarra, dating back to the early days of the NSWRL. Perhaps it's a coincidence but the Illawarra RL comp began in 1911, the same year that St George juniors were formed and the district starting competing in the NSWRL third grade competition.