high ESR in a cap is a catch-22. when the cap begins drying out, the ESR goes up. when the ESR goes up, the internal temperature of the cap goes up, which causes the cap to dry out more..... you see where this is going??? eventually it gets to the point where all 3 effects follow each other in rapid succession and "POOF" it lets out the magic smoke. then you have an open (in some cases it shorts too...) cap and no filtering (or a shorted diode if the cap shorted). with no filtering, a SMPS cannot regulate, so it shuts itself off. with the cap and diode shorted (the other failure mode) the SMPS keeps trying to start, senses overcurrent and shuts down. with the open cap failure you may have a running power supply, but nothing works right because there's too much noise on the rails, and the main board sends a signal to the power supply shutting it down, so the monitor may come on for a split second, then shut off or may not come on at all depending on which supply rail is noisy. with the shorted cap/diode failure, the supply may appear dead, but have a low voltage on some of the rails, it may chirp repeatedly, the power led might flash dimly (again repeatedly).
with most equipment today that uses a "soft" power switch (cpu controlled power cycle as opposed to a physical line switch), you should have a standby voltage for the cpu chip on the main board. on the SMPS, this usually requires a separate supply, and usually comes from a small switching transformer. the standby supply is usually +5V or +3.3V and should be there when the monitor is plugged in but not yet turned on (if the monitor has a physical "hard" power switch, it must be on before the standby supply is active). sometimes there may be a +6V to +12V standby supply that operates a power relay when the cpu chip sends a power-on signal.