I do not understand why exactly, but now it works (Mint 13 "Maya", based on Ubuntu 12.04 "precise").
The only thing I did differently is to allow "unsafe" updates, in a bid to get the newest Ubuntu stuff. Previously, I had only applied the "safe" updates.
Mint adds a safety level to updates (1 to 5). 4 and 5 are not marked by default. Ubuntu does not make that distinction, so I figured it was safe enough to mark them anyway.
What made me try that is that Mint 16 "Petra" (based on Ubuntu 13-10 "saucy") did work out of the box.
The next thing on my wishlist would be to add ID-card functionality to Chrome, if that is at all possible.
Thank you all for your time and your help!