The cathode emits electrons from across its entire surface. When the cathode is not fully hot some areas are hotter than others and emit more than others. So any current pulled through the tube heats those emitting areas more. Because the current flow is designed to use the entire cathode, not just portions of it, too much current is pulled from point sources and the cathode overheats in those regions. That damages the cathode by breaking down the alkali metal oxides, which also liberates gas, fractures the surface, and causes other damage. Cathodes are robust and can take a fair but of abuse before failing, but pulling that current gradually damages the tube and eventually kills it.
The internet claims cathode damage does not occur in signal tubes because transmitting tubes are rapidly destroyed from pulling current before the cathode is at operating temperature, but signal tubes take longer to degrade. The issue is temperature, but the degradation does occur, just at a slower rate.