All correct, with one detail comment to add, which people tend to forget. The de facto implicit validation of a local EASA Part.FCL to replace a FAA PPL only applies for the country which issued the license. As the EU does not carry status of a country, there is no shortcut for flying Europe.
If you are issued a Swiss EASA Part.FCL, by FAR 61.3 you are eligible to execute the rights of a FAA PPL within Switzerland on a N-reg aircraft. You are not eligible to fly cross border in the EU with the Swiss Part.FCL. So, if you want to fly the N-reg from Switzerland to Austria, you can only do this on a FAA license with FCC radio operator certificate.