What was the role of Indian princes and aristocratic landowners under British colonial rule?

I would not characterize the situation the way Mises did.

The English hardly needed the cooperation of the Muslims and other Nabobs to rule the country. At the Battle of Plassey (1757) they demonstrated decisively their military superiority. At their option they could depose any Maharajah they wished, simply by supplying weapons and support to whatever enemies of the Maharajah they wanted to replace him. One famous case was that of Rana of Jhalawar whom the English removed when he became non-cooperative. Also, when Punjab was annexed, many maharajahs were deposed and replaced.

The English supported whatever entity seemed to be most convenient. If a maharajah was cooperative and useful, they supported him. If he created problems, or opposed them, they would seek to have him removed and replaced.

Money and commerce were the driving factors in these interactions, not ideology.

Yes. The evidence that the British relied upon the support of Princes and land-owners is to be found in profusion in the recorded speeches and official communications of

1. every single Governor General or Viceroy
2. every single Secretary of State for India

However, there were periods when the policy on the ground was to annex Princedoms without an heir (the doctrine of lapse) and also to bypass 'zamindars' (large landlords who were tax-farmers by origin) and raise revenue from the cultivator (ryotwari system). However, this weakening of the traditional Social order- especially in Oudh and with respect to certain Maratha states- was blamed for the Mutiny of 1857. Once India came under direct rule by Westminster, it was the practice of each Secretary of State or Viceroy, on taking office, to reaffirm British determination to uphold the its duties as suzerain power and to acknowledge the loyalty shown by Princes and land-lords.

It is a different matter that the British, like previous paramount powers, reserved the right to remove a Prince or zamindar on grounds of disloyalty, incompetence, moral turpitude etc. However, this was in keeping with Indian tradition. What mattered was whether the new ruler belonged to the Ruling clan. However, this policy could not be successfully extended to Afghanistan. One reason British rule became more secure, not less so, was because the Aristocracy could see that when a Princedom or Estate was taken into administration (because the heir was a minor) then the treasury was replenished and so the dynasty became more secure.

Could the British have pivoted away from the Princes and Zamindars to base their support on the middle class? That was certainly the plan behind the setting up on the Indian National Congress. But it failed. The merchant class supported lawyer-politicians who steadily became more radical.

During the Second World War, after Hitler's invasion of the USSR when the Communists switched side, there was some idea of by passing the middle class and seeking support from the Workers. However, this was merely a temporary expedient. In the end, the Princes and Zamindars proved unable to assert themselves against the middle class and so they lost their power. But this happened after Independence.

Bankers, businessmen, landowners, nobles and princes all gave active support to the British. Indigenous collaborators and "traitors" such as Mir Jafar were key elements the British exploited to their own advantage. They promoted and demoted officials and bureaucrats to suit their ends. These are well recorded- the active support by the Jagat Seths and other bankers, the various nobility and princes.

The history of this period is now seen as a mercantile expansion- and the role of money, the motivation for acquiring wealth are seen as important factors. So the social milieu, complex economics, dynamics of a center-periphery competition and the transition from agrarian to mercantile economy are seen as contributing much more to the equation- it is no longer viewed as a simple 'military expansion' history.

Colonial history is therefore not a simplistic ruler-subject equation. Rather the British are seen as active participants in a greater and more wide-ranging transition. (This is notwithstanding the post-colonial lens which also views the construction of subject identities and many other 'epistemological violence' such as knowledge production.) For a very good introduction that refers to many other scholarly works, see this book.