Unable to edit pages on AEM Sites console on author

Up until a day ago, editing pages was possible on our author for the users in a specific group with admin (i.e. total) privileges, so that if they went on Sites they would see something like this:

Recently instead (and I cannot figure out what changed), logging in with the same users produces the outcome below: much more limited options... and in particular, the inability to edit pages.

We are running on a 6.5 instance. All the bundles are in Status Active or Fragmented.

If I inspect the logs I find no relevant errors. The only thing I can register is that, if I forcefully try to edit a page by going on /editor.html/content/pagepath.html while logged with an admin account, similar errors occur:

The names of custom code are omitted from the images.

In terms of permissions, the users of this user group have this level of access:

What am I missing? What else is required for these users to be able to edit the pages?
Thank you all very much.