I am starting to suspect either the power supply or the motherboard at this point. Advice?
I agree with your suspects.
The computer restarting could be a corrupt BIOS setting. While normally disabled, most BIOS Menus let you set power options to "Boot when power is restored" (or some similar verbiage). This is often used on servers and unattended computers to boot up after a power outage. So you might replace the CMOS battery. This will also reset the BIOS - so immediately boot into the BIOS Setup Menu to reset date and time, then save and exit to boot [hopefully] normally.
Swapping in a known good PSU will reveal if that is the problem (assuming all power connections are currently tight and secure). I would certainly try those two options before thinking about replacing the motherboard. There are too many issues with replacing the board - not least of which is the Windows license. If using an OEM/System Builder license, you cannot "upgrade" the board - you MUST replace the board with the identical brand and model number (or suggested replacement by the
same maker, if your model is no longer in production), or plan on buying a new license.
If a full retail license, there is no problem transferring to the new board.