Why does a Red Giant typically lack the mass to proceed past helium depletion in its core?

It lacks the necessary mass to ignite carbon fusion in the core.

The evolutionary endpoint of a red giant is determined by a critical mass threshold governing internal pressures. For stars originating around the mass of the Sun (up to about 8 solar masses), after the initial hydrogen core fusion ends and the subsequent helium core fusion phase (often initiated via the helium flash) concludes, the resulting core material is insufficient. Specifically, the gravitational pressure and temperature achieved in the remnant core are simply too low to overcome the necessary energy barrier required to initiate the fusion of the next heavier element, carbon. Without the ability to start this subsequent core fusion stage, the star's active nuclear lifespan ceases, leading it to shed its outer layers and contract into a white dwarf remnant, unable to forge the heavy elements seen in more massive stars.