I installed a new (threaded) headset on my old bike: a cheapo https://www.necoparts.com/en/product/H841CW.html. The thing is impossible to adjust – I can get it to a point when it still knocks, but also rubs quite a bit. The rubbing looks concentrated in a single spot though – worst with the bars turned about 90 degrees to the right.

I've popped both cups out and pressed them back in, inspecting everything on the way, to the exact same result.

Feels like the only remaining thing is the crown race being on crooked, but I don't have means at home to take it off and reinstall it. Is there anything worth checking still before I take it to a shop?

Had been fighting a fixer upper with busted ballbearings in there today..
Those things are a "1 specific kind will fit" so I spent some time going through our bin of second hand ballbearings until I found the fitting ones.

Problem was resolved by a shop – but it took them a few days too. They diligently took everything apart and back together again, turns out everything was installed correctly, yet the rub persisted.

In the end, turned out that the top adjusting cone (part #3 on the chart) was actually crooked from the factory. They replaced it and it's now running smoothly as it should be.

Be careful with these cheap Neco headsets folks :)