The global linear stability of a swirl-stabilized laminar flame is analyzed with a monolithic approach based on linearized reactive flow equations. The computational set-up is axisymmetric with an embedded and spatially resolved swirler model to circumvent the use of ad-hoc swirl profile and fluctuation at the inlet. An input–output analysis reveals that the azimuthal and axial components of inertial waves dominate the contribution of inertial waves in gain modulation at low and high frequencies, respectively. A resolvent analysis then identifies the optimal amplification mechanism, which is found to correspond to the flame angle oscillation mechanism observed in experiments. The large gain separation explains why this optimal mechanism also appears in experiments or simulations with acoustic forcing at the inlet. Finally, flame displacement is correlated with radial velocity fluctuations at the base of the flame, which are quasi-normal to the flame sheet. This component of the fluctuating velocity is amplified along the flame sheet until it reaches its tip only if flame-flow feedback is active.