I would have thought someone would have said one of the uber expensive places like Salt & Straw or Humphry Slocombe. However, just as an old fashioned, they don't make it like that any more, I would thing the original Swensen's in San Francisco, which is the last one in the United States still using the name. They aren't really affiliated with the current company with the trademark name, which operates completely outside of the United States. And I remember when there were Swensen's shops all over my area, and where they would make the ice cream on site where you could usually see the machine.
I've enjoyed some pretty good ice cream in the LA and OC area. When the kiddo and I went on a few trips, we each researched places to get ice cream. In LA there was Mashti Malone's in Hollywood. Apparently it's operated by immigrants from Iran where it's more a Persian style ice cream with flavors like rose water. The name was one of the brothers who started the business and Malone was already on the sign of the previous tenant. So the name came from reusing the sign to save money. I've had some interesting ones like cherry. On another trip we tried a different Persian ice cream shop called Saffron & Rose in Westwood.
There's another one in the OC area called Hans Homemade. We've only been to the one at the Anaheim Packing House. Really good, although my kid was disappointed they didn't have licorice, but settled for ube. My reading of their history is that they use the same basic recipe from Swensen's where their founder was a former franchisee.
As far as other places go, I've heard of the Berkey Creamery at Penn State. Supposedly really good and they use it as a testbed for food science.
