Thank you very much. So, the permissions were okay, the files are accessible.
I have tried forcing the secondary PDF parser, to see if that makes any difference, and it looks like it did the trick. Somehow the default parser (the better one) could not decode the documents. When I switched to the secondary, it extracted 10 times more keywords. Please change this option like so: https://i.imgur.com/slNzbz9.png
After that, the secondary parser will be used with future documents, and should index them correctly.