Thanks everybody for your explanations! I have ~50 controllers that I all dissembled & cleaned, but I never performed maintenance on an arcade stick. With your explanations & this video it finally clicked:
https://www.youtube.com/watch?v=ADj9AJLwADs
I started by trying to free the stick module from the controller casing by removing the ball. I did not succeed. I can see that shaft is screwed into a thread in the ball, but I just can't break it loose. The metal of the screw head on the shaft started to fray so I gave up on that idea. No big deal, I don't need to replace the ball and I can perform maintenance without removing. I had to file down the frayed metal on the screw head a bit so all the components would actually slide of. Worked fine.
Understanding now what the point of the little metal washer was, I could now dissembled everything (I put on some protective eyewear in case of explosive springs, but no such event occured, thankfully). I cleaned all debris/dust, cleaned with Isopropyl and applied silicone grease to the bowl and the half-sphere counterpart on the stick shaft.
After re-assembly I find the stick feels slightly worse than before

After careful disassembly and double checking everything with the pictures I took, I can safely say that I reassembled the stick correctly and nothing has been damaged. Narrowing down the issue, I can tell that the problem is simply still how the stick pivots in the bowl. Even if I just have the shaft, the half-sphere bit on the stick and the bowl from the micro switch assembly, movement does not feel great. Basically, once it motion everything glides very well, but after resting the stick it takes a bit of force to get it pivoting again and you can hear an audible pop as it breaks free from the friction. The silicone spray I used seems fine and is labelled as usable with plastic parts, but it's not doing the job here for some reason. There is no dirt, debris or else that prevents fluid motion. I tried to remove and re-apply the grease, it's clearly better without. That stick-and-break-free with the audible pop is worse with the grease on.
Some more pictures of the stick in case anybody is curious:
https://imgur.com/a/VPzwx
Seems like a nice stick overall! And yes, only the JP ones look like this, the US/EU ones have rubber domes, it's just a stick mounted on a dpad, basically.
So, what do I try now? Order some better lubricant from Japan? I assume just like with the micro switches, the Japanese made sure to only sell inferior lube to us western devils :/