And then the FAA entered that little line into FAR 61.3. Which is where the confusion really starts. You can fly an N-reg within the UK with a UK-issued licence and valid medical. It doesn't say which licence, or which restrictions apply. In fact, the way I read it I can do ATPL-type stuff in an N-reg within the UK, with my UK-issued PPL. If we need further clarification on this, it's the FAA which should provide it, not the CAA.
I think there's actually very little or no confusion from the FAA perspective. They mean exactly what they say - they accept any valid license that was issued in the country of operation. ICAO does not make US Federal Law and has nothing to do with the license required by the FAA regulation.
One of the numerous advantages of the FAA regulations is that they are intentionally basic and unlike the total CF of European regulation, the FARs achieve reasonable success in being unambiguous.