Geothermal power plants are almost exclusively operated as "baseload"power, generating at their maximum rated output at all times. However, as variable renewable energy sources see greater deployment in energy markets, baseload is becoming increasingly less competitive relative to flexible, dispatchable generation. Herein, we employ reservoir simulations and optimization modeling to investigate the potential for EGS power plants to adapt to this new market paradigm by providing flexible generation and energy storage services. A novel geothermal system design is considered whereby energy is stored as pressure within an engineered geothermal reservoir, and is used to drive greater production flow than would otherwise be achievable during periods of high electricity demand. Based on multi-physics reservoir simulations, we develop a linear optimization model that captures the transient pressure and flow behaviors within the geothermal reservoir. We use this model to optimize the investment decisions and hourly operations of a first-of-a-kind flexible geothermal plant against a set of historical and modeled future electricity price series. We find that operational flexibility and energy storage can provide significant benefits for a geothermal plant in a market with high electricity price volatility, with revenue improvements of 4%-52% compared to a baseload plant operating on the same price series. Surface generating facilities are oversized by up to 41% in order to take advantage of the greater flow rates provided by reservoir pressurization. Sensitivity analysis assesses the variation in outcomes across a range of subsurface conditions and cost scenarios.