After working for over 4 years, a custom PC won't POST

The motherboard has suffered a catastrophic failure.

This was always the most likely possibility. The description of the CPU fan as rhythmically starting and stopping by itself almost always points to a failed motherboard (or one that needs a CMOS reset, which has already been ruled out.)

The failure rate for motherboards compared to CPUs is also vastly higher in general. I have had to replace over a dozen motherboards but have yet to come across a single CPU failure. CPU failures are rare as they have fewer potential points of failure than motherboards and almost always outlive them.

In terms of proof that it's the motherboard and not the CPU, the ASUS Z-97 manual actually describes how a failed CPU can easily be recognized by the illumination of a dedicated CPU POST-failure LED on the motherboard itself.

The manual also describes how if the motherboard does find an error with either the CPU, boot device, DRAM or VGA module, it will not reboot itself as is happening in this situation, but will simply keep the corresponding POST-failure LED steadily lit until the error is resolved. That's not what's happening here: it's continually restarting itself.

Everything points to a bad motherboard.