The integration of a decentralized home energy management system (HEMS) marks a pivotal advancement in the pursuit of enhanced energy efficiency and sustainability within modern households. While numerous studies focus on developing efficient and innovative programs from technical perspectives, the willingness of individuals to adopt these systems is equally crucial for achieving widespread adoption and ultimately creating a more energy-efficient society. Based on a large-scale online survey with 1,196 participants in California in 2021, we investigated the intentions related to the adoption of decentralized HEMS, particularly considering demand flexibility encompassing both air-conditioning (AC) and electric vehicles (EV) control, specifically focusing on socio-demographic disparities. Our analysis found greater openness for allowing HEMS to control AC usage compared to scheduling EV charging, possibly due to immediate comfort needs or trust in AC predictability. Low-income households showed less flexibility in adjusting both AC and EVs, while high-income households were less likely to decrease EV charging. Furthermore, homeowners exhibited greater flexibility compared to renters. Disparities between different racial backgrounds in EV charging time-shifting were more pronounced than in AC aspects. Our findings indicated that vulnerable populations may lack the flexibility and resources necessary to shift their energy consumption patterns effectively, potentially amplifying energy-related disparities and exacerbating their energy burden. Policy recommendations highlight the need for multifaceted approaches in addressing demand flexibility and energy management, especially with emerging technologies like EVs, to ensure equitable strategies for promoting sustainable energy practices across diverse communities.