When I was working on a cruise ship, a noticeboard poster said it all: “Please remember that Americans consider it acceptable to brag about their kids. Australians do not.”
I often see people talk about "tall-poppy syndrome" as an innately negative thing. Personally, I see the phrase itself as entirely neutral, and I view the cultural trait it represents as a mostly good thing. The article claims "ambitious actors have to go to Hollywood to become rich and famous", but I don't think that's because of Australian culture. It's because of the size of the American economy and the scale of Hollywood.
Look at the most famous Australian actors, even the ones in America, and they're a far cry from the egotist that is the stereotype of an American actor—though America also has its fair share of "down to earth" celebrities too. I don't think our culture stops us from succeeding at all, it just stops us being arses about it, or succeeding in as many predatory ways.