The power, status and stability of the Habsburg Monarchy rested on its ability to act autonomously from the other states of the Holy Roman Empire, whilst simultaneously remaining the most important dynasty within the Reich’s borders. This was done by remaining a dynasty which identified itself as the rulers of a state which gave it legal justification to be involved in the politics of the Holy Roman Empire, in this case Austria. The lands of Austria also contained the nobility who consistently acknowledge the right of the Habsburgs to rule as sovereigns and were the most co-operative groups in the Estates of the Habsburg realms. As such, unlike Bohemia and Hungary, the Austrian nobility and the connections they had with the dynasty provided the Habsburgs with the power it needed to establish its primacy in the area. In political layman’s terms: Austria was to the Habsburgs what California is to the Democrats in the United States Presidential Election – a banker.

The right to rule as a monarch was not as absolute as we believe it to be. Sovereignty and the power to maintain state control, as well as defend an areas system of governance, relied upon the interaction between the Habsburgs as titular figureheads of their respective kingdoms and duchies, and the local nobility who directly oversaw taxation and military recruitment. Despite what early twentieth century historians argued, this marriage of sovereignty and local authority existed well into the nineteenth century.

Reforms of the mid eighteenth century undertaken by Maria Theresa and Joseph II attempted to curtail the power of Local Estates. They were effective in creating a fiscal-military system which checked Prussia’s sphere of influence and allowed the Monarchy to participate in a series of alliances as equal or nearly equal parties. However, both rulers were unable to put in to place an effective bureaucracy which efficiently maintained complete control over taxation, legislation and local governance.

The Estates were slowly able to harness more authority, even after Joseph II progressive reforms and attempts at state centralisation, until by the beginning of the nineteenth century had effectively the same autonomy as they did pre-1750. The standing army, though, was still a powerful symbol of Habsburg authority now under their direct control.

What does this mean: The assumption that the Habsburg Monarchy had the same central authority as other modern nations is incorrect. However, by 1800 they did possess the acknowledged right to rule as sovereigns in a monarchical government. This was a significant effort, especially when considering the internal pressures from Hungary at the beginning of the eighteenth century and the continual threat from France, Prussia and the Ottoman Empire.

To answer your Question as to why Vienna:

Hungary lay outside the borders of the Holy Roman Empire, was far less centralised and relatively poor in tradable goods, agriculture and natural resources. The Kingdom also possessed a hostile nobility whose power in the Hungarian Diet (Parliament) was maintained because the Habsburgs inherited the crown through marriage. Thus, the legal system of the country had to remain the same lest the new Kings risk a rebellion.

Bohemia also maintained the same legal frameworks which necessitated the continuation of Estates. These local governments were made up of cautious Bohemian nobles hesitant to provide the Habsburgs with complete authority. The authority of the Habsburgs was challenged in 1620 by the Bohemian Revolt. After the rebels defeat at White Mountain the Habsburgs were able to govern their own kingdoms without interference. This was ratified by the Peace of Westphalia in 1648. The Habsburg Monarchy then slowly replaced the Bohemian nobles with Austrians who possessed greater loyalty to the Dynasty. Bohemia was the most resource rich territory, and most populous, but ethnic and cultural divides alienated the local populous from what was a German dynasty. Concurrenlty, the Habsburg still maintained an identity which allied themselves politically with German peoples.

More importantly, both the crowns of Hungary and Bohemia were not hereditary, and the ascension of a Habsburg ruler needed to be confirmed by the parliaments of both kingdoms.

Vienna and Austria possessed the largest base of loyal subjects, and nobility whose memory of the Habsburgs always placed the dynasty as titular rulers of their lands. The nobility of Hungary and Bohemia continuously remembered a time before the Habsburg Monarchy and were hostile to any attempts to subsume their cultural and linguistic traditions into a greater German identity. Hungary lay outside the boundaries of the Holy Roman Empire, was poor and underpopulated and comprised of a large portion of the nobility who were hostile to the Habsburgs. It was not until greater concessions were made between the Habsburg Monarchy and the Diet of Hungary, as well as significant alliances between the larges noble houses of the Kingdom, that dissent in the area was kept at a minimum. Bohemian nobles were slowly replaced after the 1620 rebellion, until Austrians made up a substantial amount of that lands parliament.

Vienna though remained a significant city, thanks to its position on the Danube, proximity to the Italian peninsula, its centrality to the Habsburg loyal heartlands and the Dynasty’s affiliation as the first German family. Up until 1866, the city was the German city of the early modern and modern period. The German speaking people’s reorientation to Berlin did not take place until 1871. Some may argue this reorientation has never taken place.

To add a slightly different way to explain what /u/ErzherzogKarl said in his post.

The over-arching long-term goal of the Habsburgs was to control the HRE emperorship, and to continue their attempt to assert centralization and control over the HRE.

Until the 1520s, they had to rely only on the resources available from their hereditary lands in Austria. Thus, in the 1490s Maximilian I attempted to push an imperial reform that would give him greater executive power. That reform went in a different direction than he intended it to, but the dream of asserting further control on the HRE stayed alive.

However, as I wrote here, his successors had to choose between Spain and Germany, with Charles V being the only one who was to become both the King of the Spains, and the Emperor of the HRE. It can be said that his early ambition of keeping both -- for himself and his son -- hurt the long-term prospects of the Habsburgs in Germany.

In particular, in an era where personal presence mattered a great deal, the fact that he was on the road away from Germany for long periods of time meant that his brother Ferdinand became the go-to ruler that the princes of the HRE preferred. At the same time, by fortune, Ferdinand gained the crowns of Bohemia and Hungary. The settlement after Charles V's abdication meant that Ferdinand's line now held the HRE Emperorship, and the titles in Austria, Bohemia, and Hungary.

But not all was well. Austria was yet to be subjected to counter-Reformation. Bohemians were a fragmented bunch, the estates split not only along old feudal lines, but also along religious enmity. Hungary was subjected to Ottoman attacks, having lost a major part of her territory. But between Bohemia and Hungary we can draw some contrasts. The challenges the Habsburgs faced in Bohemia were largely internal, while in Hungary it was external. Thus, in Bohemia they had to play a political game, while in Hungary they used the external factor -- the threat of Ottomans -- to force unity of its aristocracy to preserve their domains. In fact, this same threat was also used within the HRE to collect "defense against Ottomans tax" that continued to be paid even in the worst days of the Thirty Years' War.

Now, instead of vying for further centralization in the HRE, and incorporation of the new domains into HRE structures, the Habsburgs started to pursue the opposite strategy of keeping the HRE structures away from Bohemia and Hungary. As the Habsburgs doubted their control over the HRE, the new crowns were incorporated outside the kreiss system so that the Habsburgs control over them would not be subjected to too much imperial oversight, especially when it comes to justice.

Thus, Bohemia and Hungary were used as means to empower the Habsburgs in the HRE through various ways.

To give a very succinct example, when Bohemians revolted against Ferdinand I in 1527, merely a year after he assumed the crown of Bohemia, Ferdinand asked for assistance from the HRE. But the HRE princes feared suppression of heretics there would empower Ferdinand to do the same elsewhere in the HRE, so they refused this request under the grounds that Bohemia was a foreign land, even if its king had electoral power in the HRE. In response, Ferdinand and his successors used this argument to keep HRE meddling in affairs of the crown of Bohemia.

All this is not to say that the strategy was consistent throughout history. Rudolf II favored Prague so much that he set his seat of power there, and made deals with heretics in Bohemia through the infamous Letter of Majesty so that he could gain their support in his "crusade" against the Ottomans. All this led to the throwing of Habsburg administrators off some window in Prague when his successor tried to re-assert power.

After all of that, is it a surprise the Habsburgs set their capital in Vienna?