The chip is doing precisely what it's supposed to be doing:
During startup, there is initially no supply voltage to the chip.
Current is supplied from a chain of resistors from the rectified mains. This charges the smaller filter cap.
When supply is high enough (usually 14-18V), the chip wakes up and starts driving the transistor. As it goes, it's drawing more current (usually 1-20mA) from its supply, which is more than the charge current, so the supply starts pulling down.
If the chip isn't able to supply itself, it shuts down as the voltage drops out, usually around 8-12V.
If the chip turns off, supply current goes low again, and the filter cap resumes charging.
This is what you're hearing. The transformer clicks audibly (due to physics of the core material, or loose windings), then it turns off for a while before it goes again.
When you power the chip externally, it keeps operating, so you don't hear clicking. It's just running flat out, all the time.
In normal operation, the chip's supply droops down during the first powerup cycle, while delivering enough power to get all the voltages (coming from the transformer) up to operating level. Which includes the auxiliary output on the mains side, so that the chip can be self-powered. If all the outputs come up to normal voltages, then the aux side will as well, and the chip stays running. If one or more are kept low (due to excessive capacitance, or a heavy load, or a short circuit), it will give up, wait, and try again, maybe succeeding after a few tries (each time, ratcheting up all the voltages, until it can keep itself operating).
So your problem is absolutely a classic brown-out or short symptom. One or more of the output voltages has failed, somehow. It could be a shorted transformer (as mentioned: likely to take out the switching transistor as well -- but not necessarily guaranteed), shorted diodes, capacitors, loose wires, failed resistors (I suppose it's possible that a load resistor has failed with a low resistance), who knows. You will find a low resistance somewhere; it might not be obvious (a normal transformer still measures ~0 ohms at DC), or it might be something dumb and tricky (a capacitor that's failed in such a way that it acts like a zener diode, only presenting a short at higher voltages??), but it's something. Keep looking.

Tim