Thank you for all the help. I found out what was the problem:
My clutch has two extra rings, that are placed inside, closer to the engine. The set of 8 friction plates and 7 discs go "on top" (from inside the bike to outside) of these. One of the rings is narrow and conical, and the
first friction plate has a wider inner diameter than the rest, so that it clears that ring. This short video (in spanish) shows the issue:
https://www.youtube.com/watch?v=dv5Gxkw4Atk(note that the placement order of the two rings is wrong in the video, but you get the idea).
Apparently recent Barnett clutches come with a different plate (the green one in the video) that also has a wider inner diameter. But unfortunately my set did not come with that. Either I leave out the conical ring, or I find the special first plate.
Since the original clutch was not too worn at all (all plates seemed identical and still with plenty of friction area), I decided to fit it back in. In the mean time I will look for a Barnett first plate. Any suggestions on how I can get one?
So after 3 days of work (last Sunday, yesterday and today) and spending £300 (clutch, tools, bolts and gasket), I now have everything exactly as it was when I started.
Plus points are:
* The oil chain is now tightened.
* I know that the clutch disks and plates are ok. With 30k miles, it means that the lubrication is ok.
* I know that I have the good quality basket bearing. Thanks to @stevel for the suggestion to check that.
* I know that my clutch basket springs are loose. That is probably where most of the rattling comes from.
* I know that I have fresh hydraulic fluid.
* I can remove and assemble the clutch in less than two hours.
However, the biting point is still with the lever very close to the grip. Still have to work on that.
A big thanks to everyone for the help. This is not over, I am now looking for that extra plate from Barnett.