Implementations set forth herein relate to management of casting requests and user inputs at a rechargeable device, which provides access to an automated assistant and is capable of rendering data that is cast from a separate device. Casting requests can be handled by the rechargeable device despite a device SoC of the rechargeable device operating in a sleep mode. Furthermore, spoken utterances provided by a user for invoking the automated assistant can also be adaptively managed by the rechargeable device in order mitigate idle power consumption by the device SoC. Such spoken utterances can be initially processed by a digital signal processor (DSP), and, based on one or more features (e.g., voice characteristic, conformity to a particular invocation phrase, etc.) of the spoken utterance, the device SoC can be initialized for an amount of time that is selected based on the features of the spoken utterance.