I don't think that really makes sense for groups. You can turn off file syncing for My Library because you might not want a storage subscription and might sync files another way, and you can turn it off for all groups because you might, say, be on a network where you don't want to sync any files. But if you put attachments in a group library, they should be available at least in the online library, even if people set their own clients to download files "as needed" instead of "at sync time". We try to minimize the possibility for things that cause confusion or end up just looking like bugs in Zotero.

Note that full-text searching is designed to work regardless of whether you download files to your local client. That's the "Sync full text content" option in the Sync preferences.

The issue with the upload is true. I was indeed thinking more of an option concerning the download - exactly as sketched in your addendum.

Apart from full-text search, for topics that we work on directly, my colleague and I like to have all the PDFs locally "at sync time" (e.g. on my laptop when I should be offline) and not only "as needed". But we have several large groups that serve more as a lookup database (e.g. all publications from our institute) or topics that we are involved in less frequently, where downloading the PDFs would fill up the HD unnecessarily. At the moment, I do not sync them - but I would like to have access to the metadata and "as needed" access to PDFs also in the client and not only online.

(Let me add that, despite a number of wishes that come up during use, we are very happy of our move to Zotero and its range of functionalities - collecting literature is fun again!)