I am not surprised one bit about your personal experience with the U.S. CDC vaccination card in your "small mountain area of Tuscany" I am just a bit more surprised about your experience at the Pisa IKEA coffee bar.
Readers should remember that the vast majority of Italian restauranteurs and bar proprietors have never seen a CDC card before, while personnel at major Italian museums in large tourist cities, and the restaurants/bars which cater to U.S. tourists, are thoroughly familiar with the card, as they see many every day.
I too had the experience of showing my CDC card to someone in Florence who had never seen one before. She looked at it with great interest and curiosity, and remarked that it was the first such time for her. She did not hesitate accepting it, but did ask to see my photo ID.
Non E.U. residents cannot obtain an Italian "green pass", and must instead show their vaccination documentation (preferably in paper format). The only exception (so far) seems to be ITALIAN CITIZENS who reside outside of the E.U. who return to Italy for a (long-term) stay (or to acquire residency). There is a procedure for such Italian Citizens to apply for and obtain an Italian green pass in these cases. In my opinion, even Italian Citizens who reside outside of the E.U. will not waste their time seeking to obtain the Italian green pass if they return to Italy for only a short period of time (less than one or two months).