Why does the items disappear in the playlists (using the web app)

Items of which the files were not available/inaccessible during the latest library update are considered ‘trashed’.
If this option is activated, the items get immediately removed from the library - and from all playlists too.
Therefore you should never activate it if your media storage is outside of the server machine. There is too much which can go wrong with external media storage and cause the files to be (temporarily) unavailable.
Upon next library update, the files get found again, but of course they are not automatically added back to any playlists.