Austria started as a march of the HRE within Bavaria until it was raised to a Duchy in its own right by Frederick Barbarossa. The Babenbergs were a faithful but not a major major power within the HRE as dukes of Austria.
Then the Habsburgs, a swiss/swabian house received the Duchies of Austria & Styria and immediately were elected Kings of Germany. And from then on (mostly) they were able to secure the election of king and eventually emperor.
So was Austria (and Styria) densely populated? Full of gold & silver? Connected to important trade routes through the Danube or Alps?
Or was Austria just the largest single territory most prestigious (hereditary) title the Habsburgs held so everything eventually was put under that umbrella when they created the Archduchy of Austria?
Austria wasn't even close to the most important territory held by the Habsburgs during the period in which the dynasty became extremely powerful. Rather, what was instrumental in securing long-term Habsburg control of the Holy Roman Empire was the accession of Charles I as King of Spain in 1516. Charles was a Habsburg -- the grandson of Emperor Maximilian I and son of King Philip of Castile, who became King of Spain by marrying Joanna, a daughter of Ferdinand and Isabella. Upon his father's death, Charles also inherited his father's lands in the Low Countries. Three years later, when Maximilan died, Charles inherited the Archduchy of Austria, and six months later, he was elected Holy Roman Emperor.
Therefore, as of 1519, the Habsburgs controlled the world's largest empire and two of Europe's most important monarchies with close ties to the Vatican. As a result, the Habsburgs could lay claim to control over the HRE throne because of their influence both within the HRE and on the Vatican. This control was strengthened further by some Protestant areas of the HRE gaining independence, as well as the outward hostility that characterized the French monarchy's relationship with the Vatican beginning under Louis XIV.
Suggested reading
* The Habsburg Empire: A New History, by Pieter Judson
* The Habsburgs: To Rule the World, by Martyn Rady
* Emperor: A New Life of Charles V by Geoffrey Parker