It was not possible (12 hrs of Google searching etc.) to open the damaged document which resulted in “Can’t open the illustration”
The cause of the error was a corrupted font that resulted in un-openable Illustrator documents.
When I deactivated all user-installed fonts, I was again able to open and save documents without any errors.
To keep working, I had to delete the font and install a clean copy in the Font Book app.
The font in question was Montserrat Regular.