Why does the mass-luminosity relationship break down when a star becomes a red giant?

Its internal structure rearranges, and luminosity increases due to enormous surface area, despite no significant mass change.

The $L \propto M^\alpha$ track is tied specifically to the hydrostatic equilibrium maintained during core hydrogen fusion on the main sequence. When a star exhausts this fuel and transitions into a red giant phase, its internal structure undergoes a profound rearrangement. The star swells dramatically, causing its surface temperature to cool, yet its overall luminosity increases substantially because of the resulting enormous surface area. Since this high luminosity is now driven by structural expansion rather than core mass-dependent fusion rates, the relationship with the original mass no longer holds true; a red giant can easily outshine a main-sequence star that actually possesses greater mass.