I use Claws Mail for both my regular E-Mail and my Mailinglists as well.
As you can see in the following image it allows horizontal scrolling:

This image also shows the embedded body viewer on the bottom right. If you (like me) prefer to have your messages open in a separate window: Thats not a problem and works out of the box by double clicking them.
It is also a splendid Mailhandler with a working search engine that crawls through my 100k+ Mails in about 5 Seconds. A feature that I found very useful is that it doesn't display all results in one flat pane, but highlights the folders where it found something and then displays the mails only for that folder when you select it.
That way you not only have filter on your search elements but on the reference topic that comes with your folder structure.
You can filter Mail on arrival by the targeted address (or any other thing, quite a mighty filtering system it has) and sort it into folders by that criteria to ensure elements for one ML only lands there.
It has one downside thats next to unimportant for me but may be bothering you: The filters are applied top to bottom in a list, so if you filter for "TARGET X" first and then "TARGET Y" and receive an email that has both (and no other criteria is applicable) than one of the filter will win over the other. So if you receive Mail that gets cross posted to different Mailinglists AND you use the same E-Mail Address on both AND there is no Header info that can distinguish them consistently you will get this E-Mail in one folder twice and in the other its missing. I'm certain that you can filter it out through the server that send it, but keep it in mind. - If you only want to read MLs just create an own receiver for each of them and filter on that.
Claws Mail allows folding, ignoring, selecting, moving etc. on threads and if you happen to have users on some lists that you never want to read AND don't care about their threads either you can create a filter rule that ignores that user AND the whole thread underneath them. (By ignore I mean that the thread is still visible in the list, but will be marked as read and unimportant automatically)
The configuration of Claws has its own design idea and needs some getting-used-to.
Image taken from this site which has some further infos on Claws Mail togther with recommended plugins, guides to encryption in Claws etc. might be a bit outdated, though.
Your dots:
- I want to keep every list in its own folder (or whichever concept the new client uses). YES
- Use thread view. YES
- Allow folding threads. YES
- Show the full thread (e.g., allow scrolling horizontally). YES
Nice to have:
- Allow also writing/answering mails. YES, with templates and stuff, but not WYSIWYG HTML Mail.
- Detect when mails are CC’ed to several lists and mark them read in all lists as soon as I have read it in one of them. No, not that I know of, but I haven't been looking
- Allow saving each mailing list’s web URL and password (for changing settings). Kinda Mail Account Data is stored. But the occational Mail from Mailman that tells you your ML password can't be structurally stored (you can have an own folder for that)
- Allow (manually) moving mails into correct threads. Kinda You can alter the Mail manually as it is stored plain on HD, but thats not what you want to do. I haven't tried anything else myself.
Formal requirements
- It must be FLOSS (Free/Libre Open Source Software). YES
- It can be a local client[...], it must run natively on GNU/Linux. YES