Deep learning-based hybrid iterative methods (DL-HIMs) promise accelerated convergence by combining the complementary spectral biases of classical numerical solvers and neural operators. However, these methods frequently stall at "false fixed points", where neural updates vanish despite unacceptably large physical residuals, raising significant concerns about their reliability in scientific computing. In this talk, we demonstrate that DL-HIM performance is higly sensitive to training paradigms and update strategies, independent of the underlying neural architecture. Through a detailed analysis of a DeepONet-based solver (HINTS) and an FFT-based Fourier neural solver (FNS), we demonstrate how misaligned training objectives allow physical residuals to persist. Furthermore, we show that classical Anderson acceleration (AA) is poorly suited for nonlinear neural operators. To resolve this, we introduce Physics-Aware Anderson Acceleration (PA-AA), which minimizes the physical residual rather than the fixed-point update. Numerical experiments confirm that PA-AA successfully circumvents false fixed points, restoring reliable convergence in substantially fewer iterations. We conclude that reliability depends not only on architectures, but on physically informed training and iteration design.