A long term solution was however needed. I had heard good things from this no-bullshit-geeky Swiss company called PC Engines GmbH. They make a board called APU2E0, among other things. It is a AMD GX-412TC-based SOC product with 2 Intel i211AT NICs. And it does not have regular monitor connections. Instead, there is a DB9 serial port for installation purposes.
The board’s sister version is pictured below for reference.