The growing platformization of health has spurred new avenues for healthcare access and reinvigorated telemedicine as a viable pathway to care. Telemedicine adoption during the COVID-19 pandemic has surfaced barriers to patient-centered care that call for attention. Our work extends current Human-Computer Interaction (HCI) research on telemedicine and the challenges to remote care, and investigates the scope for enhancing remote care seeking and provision through telemedicine workflows involving intermediation. Our study, focused on the urban Indian context, involved providing doctors with videos of remote clinical examinations to aid in telemedicine. We present a qualitative evaluation of this modified telemedicine experience, highlighting how workflows involving intermediation could bridge existing gaps in telemedicine, and how their acceptance among doctors could shift interaction dynamics between doctors and patients. We conclude by discussing the implications of such telemedicine workflows on patient-centered care and the future of care work.