Enhanced geothermal systems (EGS) are an emerging energy technology with the potential to significantly expand the viable resource base for geothermal power generation. Although EGS has traditionally been envisioned as a 'baseload' resource, flexible operation of EGS wellfields could allow these plants to provide load-following generation and long-duration energy storage. In this work we evaluate the impact of operational flexibility on the long-run system value and deployment potential of EGS power in the western United States. We find that load-following generation and in-reservoir energy storage enhance the role of EGS power in least-cost decarbonized electricity systems, significantly increasing optimal geothermal penetration and reducing bulk electricity supply costs compared to systems with inflexible EGS or no EGS. Flexible geothermal plants preferentially displace the most expensive competing resources by shifting their generation on diurnal and seasonal timescales, with round-trip energy storage efficiencies of 59-93%. Benefits of EGS flexibility are robust across a range of electricity market and geothermal technology development scenarios. This is a manuscript accepted for publication in Nature Energy. An updated version will be posted with a link to the official article after publication.