Can't decide! French countryside.

Have you considered the RS My Way France tour ? My daughter absolutely loved her solo trip on that tour. Her description of her time resembles your wishlist.

No, I hadn't. Is it free to look at the itinarary?

https://www.ricksteves.com/tours/france/my-way-france-vacation

However this is longer than your proposed week and involves considerable travel. I notice a similar post of yours from last year. Were none of the suggestions there of any interest?

Apparently, your vacation plans for France keep getting pushed back year after year...

There are no school breaks in late May or early June. The only public holiday will be Pentecost Monday on May 17. So, from a "French people on vacation" standpoint, it should be pretty quiet.

There are festivals and events pretty much year-round in every region of France. You’ll just need to look into it based on where you end up going.

Speaking of locations, Annecy gets about 3 million visitors a year. It’s not really the right destination for "minimal tourism," nor is it exactly "peace and quiet" (at least, not by my standards)

Relying solely on buses and trains without a car is going to make things tough if you're hoping to explore a region (again, based on my definition of the word "explore"). Buses aren't meant for tourism, they're for commuting workers and school kids. So don't expect to find hourly buses running just anywhere. If you really do your homework to find a convenient home base, you can more or less make it work. However, you'll waste a lot of time dealing with train and bus routes and schedules.

Dinan and Brittany in general could be a great option. It's not too far from Paris by train.

You should look into it while you wait for other suggestions:

https://www.brittanytourism.com/

Dinan is a city in Brittany, one of several great places to visit there. Brittany is one of my favorite areas in France. However, not having a car would, in my opinion, be disadvantageous. There are buses to get around, but it would limit your enjoyment of Brittany.

Bordeaux is a wonderful city, with a lovely cathedral, some neat museums (esp. Museum of Acquitaine), a good tram service in the city and which also goes to the airport. There are plenty of options for tours to visit the wine regions of St.Emilion and Bordeaux, plus, on your own, there is the Cite du Vin and the Musee du Vin etc du Negoce, the Bassins de Lumiere show, a charming vieux ville, excellent food options, a possible day trip to the Arcachon Bay for the seaside......really, Bordeaux would fit your desires quite well without needing a car.

I googled festivals in Bordeaux in May and June and there is a lot going one, esp. music festivals. Do check on what's there in those months.

It also is just slightly over two hours by TGV from Paris' Gare Montparnasse. A great choice IMO!

I plugged your criteria into Google, and the AI said Dinan, so you were right all along. AirBnB has a lot of great choices for places to stay there, too.

See Laurie Beth's post and all the responses about Bordeaux. "Things you love to do in Bordeaux". It's just above yours.

Virtually any place in France would meet your criteria - relaxation (but with exercise),delicious food, beautiful gardens, lovely scenery,
ancient sites/history, minimal tourism (this one is a bit harder !), abbeys (!), and friendly locals. With such a large family, I would suggest that you first go through the listing for gites - for lodging possibilities - https://www.gites-de-france.com/fr - and then zero in on where you would like to stay and then focus on transport logistics. (I just checked Annecy and there are some really beautiful homes).

Perhaps I missed it, but how many nights will you be in France?

There's a great inn called La Ferme de l'Oudon in Normandie that could be a great spot for your family. It has a lovely indoor pool and so many animals. Quite close to Lisieux, Caen, Deauville/Trouville. The bed and breakfast itself is a destination. We went about 3 years ago and are still talking about it.