I wish I knew a town like you describe in the US, but, I'm afraid, that's just not how the US is set up. And if the place is like you're looking for, it's going to be an absolute tourist trap, all smoke and mirrors.
I wouldn't move to TN. I went to school in TN and I go back a fair bit. It's beautiful and bucolic and the politics are just atrocious. Great to visit though. Franklin is a tourist trap and the property prices are through the roof while services aren't keeping up. It has grown in leaps and bounds recently, but at its heart Nashville still is a small provincial town. It is quite insular, there aren't a lot of permanent transplants (lots of second/third/fifteenth homes of celebrities though) it's not a very European feel. People are very nice but it's the south. Expect to be asked which church you attend in a casual conversation on the regular. The airport is also not super great, not a terrible one, just small and not many international flights at all.
I love the Carolinas. It's also the south but Charlotte feels a lot more cosmopolitan than Nashville does. There are quite a few beautiful smaller towns and suburbs that might suit though and, overall, NC will have a bit less of the Bible-thumping crowd than TN (although don't be fooled, there will still be a lot). I saw Davidson mentioned above - it's beautiful and quite educated because of the Davidson College. It is also right by Lake Norman which is beautiful (and $$$). I wouldn't say the Main Street is quite like High Street but there are enough restaurants to maybe eat in a different one every few days for a while. And Charlotte airport is better than Nashville because it is a hub for one of the legacy carriers.
Honestly, instead of looking at midsized free-standing cities, I think you might be better off looking at the RIGHT suburb of the RIGHT big metroplex. The million dollar question would be which one that is, of course. I personally think the East coast is beautiful but I haven't ever lived there long term so I can't give very good advice about specific towns. I'm sure there are places in CT, MA and MD that could fit the bill of what you want. I know there are several places like that in VA, mentioned above. West Coast scares me because it seems there's always some kind of natural calamity befalling their cities and I'm the nervous kind, not fond of wildfires or earthquakes so I haven't ever considered that region.
I live in Midwest now, in Chicago area. It's not perfect but there is much to love about this area. Something like Barrington, IL https://www.barrington-il.gov/about/ or Geneva, IL https://www.geneva.il.us/ might appeal to you. Both are small, distinct towns that are actually Chicago suburbs so you get the best of both worlds - small town feel with all of the big town amenities, and not quite the crazy price tag of places on Chicago's North Shore (home to some of the most expensive zip codes in the US). Barrington is about 30min from O'Hare and Geneva is less than an hour. Both places have good k-12 schools and our state university is well rated (although I'd recommend sending the kids back to Europe for Uni because the prices here in the US are shocking even for state schools and stroke inducing for privates). The drawback is weather and some people would say the blue state taxes are oppressive. Taxes are considerably less than in Europe though (healthcare and education are out of pocket but thats everywhere in the US), winters aren't the greatest and the summers can be hot and humid.
My dream place would be just like you describe + somewhere on the ocean. Until I find it, I'll just stay in Chicago area for the kids education (blue state means our k-12 schools are MUCH better than those in TN and even NC although NC has wonderful universities).