For both historical and financial reasons, American Express acceptance in France is quite low.
Visa and MasterCard were associated from the start with the CB program, so anywhere they take cards they will take Visa and MasterCard, it's a standard part of the contract a merchant would get from their bank for card acceptance.
On the other hand, American Express acceptance often needs to be requested separately, and the fees are often much higher, so merchants often don't bother.
American Express cards will be accepted in most upscale hotels, restaurants and shops (mostly those that cater to high-end tourists), but other than that, it's pretty much a lottery.
American Express have been working hard to expand acceptance, so you can have some nice surprises. They are actually accepted in stores like Carrefour (though not the very small ones) or Monoprix.
They have a site where you can look up merchants which will accept AmEx: https://www.americanexpress.com/fr-FR/maps
As a side note, even if the stores accepts contactless payments and American Express, it's nearly impossible to find a place where you can use American Express with contactless or Apple Pay, the terminals are usually not configured correctly for this.
You will most likely need to have a Visa or MasterCard card, or use cash, in most places. Note also that ATMs that allow you to make withdrawals (cash advances) on American Express are quite rare as well.