We study numerically and theoretically the gravity-driven flow of a viscous liquid film coating the inner side of a horizontal cylindrical tube and surrounding a shear-free dynamically inert gaseous core. The liquid-gas interface is prone to the Rayleigh-Plateau and Rayleigh-Taylor instabilities. Here we focus on the limit of low and intermediate Bond numbers, Bo, where the capillary and gravitational forces are comparable and the Rayleigh-Taylor instability is known to be suppressed. We first study the evolution of the axially invariant draining flow, initiating from a uniform film thickness until reaching a quasistatic regime as the bubble approaches the upper tube wall. We then investigate the flow's linear stability within two frameworks: frozen time-frame (quasisteady) stability analysis and transient growth analysis. We explore the effect of the surface tension (Bo) and inertia (measured by the Ohnesorge number, Oh) on the flow and its stability. The linear stability analysis suggests that the interface deformation at large Bo results in the suppression of the Rayleigh-Plateau instability in the asymptotic long-time limit. Furthermore, the transient growth analysis suggests that the initial flow evolution does not lead to any considerable additional amplification of initial interface perturbations, a posteriori rationalizing the quasisteady assumption. The present study yields a satisfactory prediction of the stabilization threshold found experimentally by Duclaux et al. [J. Fluid Mech. 556, 217 (2006)].