In my home one distribution board indicate supply but when I connect any appliance to board the board indicator is off and appliance does not work. Other distribution boards are working.
You probably have a wire break, a corroded or loose terminal, broken receptacle, or something somewhere. All of those situations can create a high-resistance on the circuit. That resistance can be low enough that a low current device (such as an indicator light) can still work and a multimeter will measure the correct voltage. But a high current device won't work.