Oh man, something I can somewhat effectively contribute to, with sources I have right next to me.

So, the first thing is the wikipedia link you had is not the right organ. That's the Reichshofrat which was part of the Holy Roman Imperial legal system, focused primarily on providing advice to the Emperor in administrative, legal, and other affairs. This body was usually considered to be dominated by the Emperor, compared to the Reichskammergericht which was more dominated by the Imperial Estates. The body you're looking for in the military role is the Hofkriegsrat.

The Hofkriegsrat was established in the mid-16th century as a central military authority for purposes of administrating the military. The Hofkriegsrat was the highest legal authority, and was responsible for facilitating communications between the Emperor and his commanders in the field, and even diplomacy with the Ottomans until 1753. The establishment of a standing Austrian army in 1649 made the Hofkriegsrat much more important, and much larger. The Hofkriegsrat technically had control over the General War Commissariat, which was the main body for supplying and funding war, and it was often presided over by some of the Empire's top military men (e.g. Montecucolli, Prince Eugen of Savoy). It generally functioned as an administrative body, presenting to the Emperor the needs of the army in terms of money and men, ensuring that money and men ended up where it needed to, etc. Of course, Habsburg Austria being Habsburg Austria, its responsibilities and duties often overlapped with other organs of the Habsburg state. In many ways, it was a quintessential example of Habsburg administration, with limited ability to actually administer, and thus prone to devolution of power to provincial estate and even private actors, and constantly concerned with the provision of money for the army and war machine. This led to a convoluted mess of responsibility and lots of bureaucracy to ensure what there was kept moving.

Unfortunately, the Hofkriegsrat is one of the less well understood organs of the Habsburg political and military administrative. The Habsburg military had a tendency to become utterly dominated by individuals (Prince Eugene in the early 18th, Archduke Charles in the Napoleonic), and these men would often have a role as President of the Hofkriegsrat. However, the person in charge would not always be the most competent of individuals, and such weakness would propagate throughout the whole of the military apparatus. In some ways, its inefficiency was bureaucratic. Its responsibilities grew, leading it to grow larger and more unwieldy, and it often had civilian leadership in positions above that of military leaders. While the Hofkriegsrat did not necessarily directly command field armies, its President and leadership was often the same as the major military leaders in the field.

However, what this often meant is that the leaders in the field would often hold bureaucratic ideas, that stressed the importance of fighting defensively (for Austria was chronically underfunded, due to limitations of its administrative structure). This mindset among the officer corps encouraged a deep cautiousness that the Austrian military was often loathe to break away from. It could be difficult for a local commander to break from the tradition encouraged from on high, even if they did wish to act more aggressively in the field.

There were definitely massive attempts to reform the Hofkriegsrat throughout the Empire's history. Maria Theresa and Kaunitz sought to streamline the Hofkriegrat in 1745, and Archduke Charles did the same during the Napoleonic Wars. Of course, such reform was difficult to push through, and it required significant political weight to actually occur. Most reforms focused on streamlining the bureaucracy, trimming the size of the Hofkriegsrat and establishing a more well defined system of administration.

In the end, it seems more likely that the Hofkriegsrat was more a symptom of the deeper issues within the Austrian military than the cause. The military was defensive minded, and progression was bought through seniority. As such, the Generals who went into northern Italy to face Napoleon in the Wars of Revolution were old and heavily tied into the bureaucracy of the military, rather than dynamic leaders in the field. Hochedlinger in particular fingers this 'crisis of leadership' as a major cause for continual Habsburg weakness in the field. The Habsburgs suffered from many limitations, in terms of their resource availability, domestic structure, geo-political position, etc. These limitations manifested themselves in the form of a conservative defensiveness that permeated the Habsburg military machine. No doubt, individual elements of the military displayed bravery, skill, and excellence, however their remained a crisis at the core. So--in my view--it was not the conservatism of the Hofkriegsrat in particular, but more long lasting and deep structural issues within the Habsburg state and military that were very difficult to erase, usually requiring the attention from the highest levels to force change.

I know I haven't answered all your questions, as I don't know enough about the military administrations of other European powers to properly compare them, and my sources are sadly short on specific examples. However, I hope this has helped. Please, feel free to ask any follow up questions, and I'll do my best to continue.

Sources

Michael Hochedlinger, Austria's Wars of Emergence: 1683-1797

Richard Basset, For God and Kaiser: The Imperial Austrian Army