score:3
The Wikipedia article lists three relevant sources on the subject:
A spreadsheet from the official EU website which indeed lists numerous exemptions for certain Annex II citizens. However there are no references to any national laws that would back up this information.
A page on the website of the French Ministry of Foreign Affairs which does indeed specify exemptions for citizens of certain countries:
Exempt from visa requirement for Franceβs territory in Europe are:
...
nationals of the following countries, whatever the reason for their stay ...
nationals of the following countries: Australia, Brazil, United States, Japan, Mexico, Singapore, South Korea, Venezuela. If you are gainfully employed, the exemption only applies if you can produce a work permit;
From this we can conclude that outside of the 8 specifically mentioned countries, all other Annex II nationals may be employed visa-free.
My conclusion is therefore that the only country where it's possible to work without a work permit is France, at least from the sources I could find.