Hydrodynamic effects are known to significantly affect the collective behavior of wet active matter systems. However, the non‐linear many‐body nature of these interactions makes them very challenging to analyze, particularly in dense suspensions. Quincke rollers are one of the canonical examples of artificial microswimmers. These are active particles that move freely above a plate due to the torque generated by a uniform DC electric field. A system involving many such particles exhibits a variety of collective dynamics, such as the disordered gas, polar liquid, and active crystal states. This work accomplishes numerical simulations of a 3D system containing many rollers in order to explicitly resolve the hydrodynamic interactions and understand the role these play in the resulting active dynamics. It reveals that the far‐field hydrodynamic effects tend to drive ordered collective motion, while the near‐field effects tend to drive disordered collective behavior, and reproduce diverse collective behavior while capturing the role of hydrodynamic effects between rollers without resorting to the approximations previously employed.