I just replaced the CPU cooler in the server, along with its (broken) mounting bracket. The old one was the stock AMD cooler, on a plastic bracket that came with the motherboard. One of the plastic hooks was broken a long time ago, so the cooler was in place, but only just.
I now installed a Cooler Master Hyper 212 EVO so hopefully the reboots are gone now. In the BIOS the CPU temperature dropped from previously 57 degrees (C) to now somewhere around 34-35 degrees. Much better.