Mostly the failure causes increased leakage, and the charging current is very small. The power supply startup voltage is pretty high so it will get at least a half dozen switching cycles in to start the auxiliary rail to actually power it before it reaches undervoltage lockout, and the IC typically in UVLO is in a very low power state. Thus the current is typically under 100uA, and this, along with the drop in capacitance, or the increased ESR dropping terminal voltage under the load of driving the SMPS power device, is what causes it to hiccup and not operate.
Thus you can have the PSU run in startup mode all the time, with a few cycles of operation every few seconds as the cap degrades, with it possibly starting after some time as the capacitor heats up from the current pulses and the heat from the startup resistors, till it eventually is not running at all or just pulsing every few seconds.