Thank you for suggesting what to consider.
I looked through TSW202 firmware changelog. There are a few changes that improve PoE operation when PoE devices are connected. However, the suspect motherboard causes TSW200 to halt the network and start blinking despite the fact that the motherboard is not a PoE device, and even when no other devices are connected. So, I suspect that there is something wrong specifically with the onboard LAN implementation on that motherboard. What might be wrong with the motherboard, however, may or may not depend on how TSW200 probes a newly connected device.
The issue happens immediately when the motherboard is powered or, if it is already powered, when the Ethernet cable is plugged in. The computer does not need to be turned on.
It does not look like TSW200 could be affected by other issues or CVEs mentioned in TSW202 firmware changelog. TSW200, while similar to TSW202, is an unmanaged switch. This fits my use case perfectly because TSW200 does not expose HTTP, SSH, or other services reducing the attack surface and relieving me from firmware maintenance and configuration administration. The wiki pages for TSW200 mention no way for the end user to flash firmware.
So, if there is a newer firmware for TSW200 to which the device is necessary to upgrade, I would need tech support. For the reference, considering TSW200 product hardware change notifications, this device was manufactured some time in July — September 2024.