Reconfigurable Intelligent Surfaces (RISs) pose as a transformative technology to revolutionize the cellular architecture of Next Generation (NextG) Radio Access Networks (RANs). Previous studies have demonstrated the capabilities of RISs in optimizing wireless propagation, achieving high spectral efficiency, and improving resource utilization. At the same time, the transition to softwarized, disaggregated, and virtualized architectures, such as those being standardized by the O-RAN ALLIANCE, enables the vision of a reconfigurable Open RAN. In this work, we aim to integrate these technologies by studying how different resource allocation policies enhance the performance of RIS-assisted Open RANs. We perform a comparative analysis among various network configurations and show how proper network optimization can enhance the performance across the Enhanced Mobile Broadband (eMBB) and Ultra Reliable and Low Latency Communications (URLLC) network slices, achieving up to ~34% throughput improvement. Furthermore, leveraging the capabilities of OpenRAN Gym, we deploy an xApp on Colosseum, the world's largest wireless system emulator with hardware-in-the-loop, to control the Base Station (BS)'s scheduling policy. Experimental results demonstrate that RIS-assisted topologies achieve high resource efficiency and low latency, regardless of the BS's scheduling policy.