You make a good point about football being played in many countries globally. However, regarding the point about continents, football is not the main ball sport in Australia. It probably ranks third or fourth most popular (after Australian football, rugby league and possibly rugby union), although Australia will be at the upcoming football (soccer) World Cup.
Australia moved from the much weaker Oceania Football Confederation, apparently in 2006, to a better Asian Football Confederation, better in the sense that the competition was stronger but not by much (if you put aside Iran, Japan and South Korea). Qualifying from this region is not nearly as tough as qualifying from other more competitive regions.
Australia as a country is known to spend real money on sports. They are able to produce top athletes, that's not a problem.
Australia has been able to participate in the Confed Cup and World Cup tournaments which helps boost its ranking because more points are allocated for participating in those tournaments.
Whether Australia poses a real threat to the dominant powers, that's a different story altogether. Moving to a Asia surely paid off dividends for them though.
But football is the only real global sport. There's a country on every continent where football is the main sport. What other sport can claim that?
So dominating football is hard, because there's loads of other countries where local tradition is to train football.