I have only been using the screen for 2 months with 2 devices. But I don't know any network and keyboard problems. Works for me as it should

I can confirm the problem with the network. I have both notebooks, which are connected to the screen, wired separately. I have the Logitech mouse and keyboard where you can connect 3 different devices together. Never any problems with it.

I myself only use the switch as a USB supplement and on macOS. Keyboard via Bluetooth.
But: Have you also tried another computer and thus ruled out that the problem is not caused by the computer (EFI system start, initialisation sequence). Since it reads like an initialisation issue. It could also be a timing problem with the system start-up. This may be why it sometimes works. Possibly upgrade the EFI of the computer or adjust the start-up behaviour (this still worked with normal BIOS).