It seems like nobody cared about that, so after running around with other weird solutions the one in the german forums actually held the solution: https://eu.forums.blizzard.com/de/wow/t/ingame-shopbrowser-bleibt-schwarz/36896
To translate the solution into english: If World of Warcraft is installed on a NTFS mounted folder (e.g. mounting an SSD as C:\Games\), the screen remains black. Unmounting it and assigning it a drive letter fixes the problem.
Also the browser writes logs into a weird directory called “…\Worl\” instead of its usual folder “…\World of Warcraft\_retail_\UTILS\”. This might also be why Mac OS users have trouble with it. Mutliple mount points is kind of a Linux thing.