It works now just fine. Cannot see any visible advantage over VNC, though, at least not at
LAN speeds.

glad things are working for you. yes, you'd only notice better speed when working remotely

There's one thing that I apparently cannot do with NX and that is attach
to an existing non-NX session, right? This would be the main purpose here,
attach to the console session from another machine.

So, I will have to use VNC for that, anyway.

not sure, I never tried it. the nomachine.com FAQ / doc pages ought to reveal pretty quickly what is possible.