Little update:
A weekend with debug logging enabled later, to catch what might cause this, it didn't happen all weekend. I turned off debug logging and sure enough, 30 minutes later it happened again. But I still got some insight/info whats happening.
From the logging in my router, it seems to happen when the system is renewing the DHCP lease. It happened 2:44am according to the system's time, and the router reports the last IP/DHCP activity for the system to be 2:44am as well.
Unplugging the Ethernet cable to "reset" everything at first does nothing, but deactivating cable networks, unplugging the Ethernet cable, exiting back to main menu, going back to enabling cable networks and then replugging the Ethernet cable finally gets it to regain a usable connection. And I suspect it will stay that way until I reboot again.
Still haven't tried the mentioned nightly yet, but will do that soon. Although, I'm not ruling out a hardware fault at this point, because the system is pretty old by now (about 10 years, it's an AM1 AMD Athlon 5350 system) so the board might be on its way out (the HDMI port is already starting to have issues when the plug is not in a certain position). Luckily, I got the exact same board as a spare that only ran maybe 2 years, so I'll test that too in case the nightly doesn't help.