Click sound could be most commonly: the hard drive, the fans on overload from overvolt.
Computer won't turn on? At all? There is a way to measure the power coming off the PSU using a volt meter. However, this is a pain. Obtaining the additional power supply will let you know, proof positive that it is in fact the PSU.
Clearly if no lights are showing up on the board itself, this is another indication it is the PSU. No HD spin-up. The only other circumstance you would get this is if the processor is shot. If the processor is dead, the motherboard will not respond to commands, even power up, from my experience.
The smell of electronics burning is so commonplace to a PSU overload that I would definitely do that first. Before you buy a replacement PSU, look on the board to see if you see any burnt out transistors. The transistors are the bubble-top looking devices that exist on the board. If even one of them is fried, the motherboard is dead. If the PSU hits a power spike, and that spike is relayed over to the rest of your components, including the motherboard, it will create a short circuit situation, or even worse, a blown out transistor. Power line overload could kill either the northbridge or southbridge of the board. It could also kill the processor. My experience over the years has been that, unless the CPU itself lost control over the motherboard and caused the overload (thus frying itself out), 9/10 the CPU is preserved while the board is dead. The CPU is somewhat protected from these types of problems.
Try and replace the PSU, but examine the board thoroughly first. For safety reasons, do not try to mess around with the power supply too much. I would say just go and get a replacement. If not, you have 30 days to bring it back. You could come to a conclusion, without the volt meter, that the motherboard is also fried
I am afraid this may be the only way to resolve the situation.
power supply in question is an Antec True Power Quattro 850W. I'm assuming a high end power supply like this should have some sort of protection, right?
No protection if you're not on a surge protector, of if the surge is so strong it runs faulty. What can also happen is the opposite of a power spike, but a drop in power, and then a compensation that creates the spike. This is even worse. This is why for high-end systems, a lot of people employ a UPS (Uninterpretable Power Supply) to filter the electricity, creating a "clean" line of current going into the system. The PSU is constantly doing a conversion from alternating to direct current, and regardless of the quality of a PSU, they are very often prone to failure. Especially if this is new.