I realize restrictions may be about to change but wanted to quickly share my experience traveling from mid- to late August, as a double-vaccinated US traveler. Switzerland and Italy both required entry forms (including for transiting by air) including information about transportation and lodging. There was some pretense about checking these when entering Italy by train from Switzerland, but forms were only sporadically checked (I wasn't checked) by border personnel. At the time I was there, no tests were required for vaccinated Americans. Sights like museums were way below normal capacity -- not because of entry limits but because of less demand. Austrian and Swiss museums, churches, and restaurants did not check vaccination cards/green passes or temperatures in my experience, but almost all indoor sights in Italy did. While it sounds like some US travelers have been able to get the Green Pass by presenting vaccination records at pharmacies in France, etc., this did not seem to be possible in Italy -- checked in multiple cities. We had to explain every time we were asked for green pass and had to present vaccination cards instead -- learn how to explain this in Italian if traveling outside major tourist destinations.
I'd also say -- we had a really wonderful time. The stricter COVID controls in Italy helped make sights safer. We ate outside. I have no idea whether this was related to a sharp drop in tourist traffic but we lucked out with church and monastery caretakers who showed us around usually locked-down areas. As long as you wear a mask (medical-grade sometimes required, sometimes not) and respect protocols (always use hand sanitizer that's provided, e.g.), you should feel welcome.