However, it looks this does not apply in Switzerland.
It would be helpful if you could tell us
why you think it doesn't apply in Switzerland.
There is an obligation under the Chicago Convention on Switzerland to accept FAA licences for flight in N-reg aircraft. There is no obligation on it to accept any other state's licence for flight in N-reg aircraft, including ones that Switzerland has issued, even though many states do accept other licences, particularly for private flights.