So no EASA medical = no EASA valid licence = no valid FAA piggyback licence
No, read the wording of Part-FCL carefully - semantics can be your friend:
An EASA Licence is valid for life (a JAR one for 5 years). A valid medical is
only required if the licence privileges are to be exercised.
Those privileges are to act as a pilot in aircraft that are registered in EASA states.
If the FAA deems a US Medical adequate to exercise EASA Licence privileges
in an N-Reg aircraft then that is nothing to do with EASA.