Self-organized dynamic patterns in dense active matter are striking manifestations of non-equilibrium physics. A prominent example is the macroscopic elliptical motion observed in quasi-two-dimensional bacterial suspensions, whose physical origin has remained unresolved. Here, we examine a minimal linear-response framework in which the fluid mediates long-range feedback between run-and-tumble bacteria. Cross-film swimming combined with Jeffery reorientation produces a phase-leading nematic response to shear. Self-sustained oscillations arise only above a critical cell density and below a cutoff film thickness. Using independently measured single-cell and hydrodynamic parameters, the theory simultaneously accounts for the observed onset density and oscillation frequency while identifying film thickness as a primary control parameter for collective oscillations in confined active fluids. More broadly, the work illustrates how known microscopic dynamics and physical interactions can be combined into a predictive framework for emergent collective behavior.