Refraining from action?

A colleague asked me to confirm the following: "Indian philosophies didn't say much about action explanation because on most of the views actions were something that we should aim to CEASE to perform". Apart from the explanation part, I wonder whether this is not an "orientalistic" oversimplification. Many of us are used to Advaita-Vedānta depictions of saints detached from bodily affairs, including actions. Conflated to that is also the Buddhist refusal of a self who performs the action (cf. the Zen saying "the arrow shoots itself"). This is certainly true, but it does not lead to the consequence that the illuminated one ceases to perform actions (since this would be impossible). Rather, he ceases to identify himself in the actions performed. In fact, the illuminated one knows that there is no agent or patient beyond the action. As for mainstream Hinduism, the idea that worldly business (or actions tout court) have to be abandoned is not as popular as many Westerners think. The Bhagavadgītā, which is possibly the most read and revered book in India, explicitly refutes the "path of non action". Every action, it says, is surrounded by error, just like fire is surrounded by smoke. Nonetheless, the solution is not not to act. Rather, one should act without interest. One should act regarding the action in itself and not its results. One should, e.g., act rightly independent of the possible negative or positive consequences for oneself. If one acts in this way, even if one eventually gains something out of the action, one is not (karmically) bound by it since the result was just the action's consequence and not its only motive. No wonder, some Vaiṣṇava thinkers liked Prabhākara's idea of obeying a prescription for its own sake.