I flew Charlotte > JFK > Milan with American Airlines. I used the Verifly App to scan my dPLF, COVID card, and test. I used the Binax proctored COVID test less than 72 hours out for my testing that was checked in CLT. Outside of that, no one on the flight from JFK got documents checked in Milan for the a CDC card, testing, or dPLF. Talked with a guy who flew in today from the US and he said nothing was ever checked at any step of the process, before flights, or at customs and he didn't do Verifly. I wouldn't use this as an excuse not to follow all the right steps, just letting you know my experience.
The hotel I am staying in did check the COVID card for allowing dining. I've tried to go to 4 restaurants with only 1 not accepting the CDC card for inside dining. Very few people are wearing masks outside but there is 100% compliance with indoor use of masks. That's my experience so far in Courmayeur.