It could be worded more clearly, but the point is that the licence itself remains valid unless suspended. revoked or expired.
To exercise the privileges then there must be a valid type/class rating and medical certificate. FCL.040 says as much. Your argument would similarly imply that the aircraft rating is also valid on the foreign licence.
Unless I see anything that explicitly says otherwise, I have no intention of getting an EASA medical to exercise the privileges of my US Airman's certificate.
I think this could go on ad nauseum, so I'll rest my case.