Can I fly to (board) Turkey without passport?

Yes, ground handling staff will want to check your documents (because airlines can be held responsible if they bring someone without proper documentation) and you will have to go through an official passport control (Schengen exit check from the French police). But as a French citizen you can indeed enter Turkey with a national ID card and you won't even have to pay for the Turkish eVisa which many other European citizens have to get.

In practice, since the rules can be very complicated, airlines use a database called Timatic to figure out visa requirements and, as of today, it does include all the information you found on the official Turkish Ministry of Foreign Affairs website so I would not expect any trouble with the airport staff.

As for the police, there is no reason they should prevent you to leave but I don't really know. Leaving the Schengen area with a valid national ID card is definitely allowed and unproblematic. I don't think they should really be concerned about Turkey's rules but I suppose you might encounter a police officer who is surprised about it and requires an explanation. Nowadays, I mostly use the automated passport gates at major airports and nobody even asks for my destination so I have no first-hand experience to report. I don't know how the machines would respond to an expired document either.