Yes, weirdly, the costs outside of Stockholm are much lower and similar to Lyon from what I can see. The tax agreements with the US for the two countries are different but possibly in ways that don’t matter until retirement. I’ve looked at Numbeo but there are so many expenses we bear now that may not exist in the EU (like summer camps that are $700 per week in our area and necessary if you work). And maybe some costs in EU we haven’t thought about. - @Ames4
There are summer camps but they will be organised with things like scouting groups. It wouldn't ever cost $700. That's just nuts.  In one sense, why bother? Instead of sending kids away, don't forget you'll work far less hours in France and have much longer holidays (the entire country leaves for August). There's opportunity to spend more quality time with them.  You can always hire an au pair.
It'll be different later when they want to go away with their friends and many schools in Europe organise things like skiing weeks. I expect it will be almost compulsory to send them skiing seeing as Lyon is right next door to the Alps. It'll cost hardly anything to send them skiing for a week.
I don't know about France but kids don't always go to school at an early age. They often start when they are 6 which is late by my country (UK) standards. And you or someone will have to be around to take them to and from school if it's too far to walk.  Many kids will go to school in the immediate vicinity of your house/apartment. They can walk there themselves once they know the way and can go in group with their friends who will live very close or next door. Â
Walking places and public transport is common in European towns/cities. Our kids (well, actual adults and close to adults) just go on the bus/tram. They've been doing it years.
BTW, here, in my adopted country, kids have to be of a certain weight (to show ability to carry books/bags) and must be vaccinated to attend school. It will likely be the same in France. Anti-vaxxers will have to do home school.
Instead of me chipping in and talking about France, a country I don't know in detail, you need someone who lives there to say something about it. I've been to France many times but I've never lived there. It's a very popular place with many people from the USA. We (well me anyway) don't see it really as foreign. It's just different to here but not unfamiliar. Everywhere is Euro-normal these days.  Brexit hasn't helped much sustain my view.
If you want to see a really bad Numbeo comparison, compare Dublin to Budapest. Madness.