To decarbonize electricity grids, CO₂ capture and renewable wind/solar are two promising pathways. However, the intermittency of these variable renewable sources and the high energy requirement of carbon capture restrict their widespread deployment. These challenges are traditionally addressed independently at the grid-level, leading to conservative costs and limited operational flexibility for both systems. Here, we examine the synergistic integration of renewables and flexible carbon capture with individual fossil power plants. Renewables provide clean energy for carbon capture, while flexible carbon capture acts as a form of energy storage to counter renewable intermittency. To assess whether the benefits obtained from integration outweigh the capital cost under spatiotemporal variability of electricity markets and renewable energy, we develop a mathematical programming-based optimization framework. We decouple the design and operational decisions in a two-stage optimization strategy to efficiently solve the large-scale problem. When applied to a nationwide case study on coal plants across the US, we observe that, for futuristic carbon tax and renewable cost scenarios, it is profitable to invest in solar-assisted carbon capture for nearly one-third of the coal plants. It reduces carbon capture cost by 8.9%, and accommodates solar intermittency while avoiding the capital cost of an equivalent battery, which is 4.4 times the solar farm cost. Furthermore, the levelized cost of electricity will be less than that of new natural gas plants with overall emission reduction between 87.5 and 91%. The integrated system thereby provides a cost-effective and sustainable measure to reduce CO₂ emissions and improve the operational flexibility of existing fossil-based systems for accelerating the clean energy transition of the global energy sector.