Another trip to the DIY store resulted in new knowledge. Even though when I bought the replacement screw set I was told there is no need to use any other sealant (silicone, for example), it turns out, there is a need to seal the whole thing. I could choose between some real strong silicone sealant used for gluing aquariums together, or silicone grease.
Since I became somewhat cautious about permanent solutions, I chose the latter. Took the whole thing apart (again), dried it out (again), and covered everything with a generous amount of silicone grease. Then put the whole thing together again.
Miracle: it does not leak. The leak has stopped. I am victorious!
The only minor annoyance now is that the bloody toilet still keeps dripping now and then, meaning there is still a leak somewhere in the mechanism.