It seems that the problem is caused by Indesign itself since it does not occur with other programs. I was able to check our internal database and I see that we are aware of this problem when saving PDF files directly with Indesign.
So when exporting to PDF, the PDF is always 'damaged' because it has 0 bytes, while the 'background tasks' in Indesign aren't showing any problems. Two of our colleagues are experiencing the same problem: when exporting a PDF from Indesign directly into Google Drive Desktop, the files for other colleagues are showing up as 0 bytes.