My (naive) understanding is that capacitors charge very quickly, but also discharge very quickly. If you want to store a lot of power and deliver it all at once

My (naive) understanding is that capacitors charge very quickly, but also discharge very quickly. If you want to store a lot of power and deliver it all at once, then capacitors are good fit. Phones, however, need a steady flow of low power, rather than a short-lived spike of high power.