The Hornec's and their direct entourage in Paris (as is the case with most of the other infamous "Traveller" crime families in France) are not "gypsies" like the traveling Roma people from Eastern Europe or the Sinti people from Western Europe, but they're described more like "Voyageurs" - indigenous traveling people that are of European descent, mostly from France, Germany or Switzerland. It's the same like in the Netherlands where the "Reizigers/Woonwagenbewoners" are mostly of indigenous Dutch or sometimes of German descent or the Irish Travellers in Ireland that are of indigenous Irish descent.

The Hornec's for instance have their origins in the Alsace region of Germany, but they'd consider themselves French more than anything. The crime families from the settled Traveller communities are probably the closest thing to indigenous "French" organized crime.
You could consider Corsican OC as indigenous "French" OC as well, but they probably don't like hearing that lol.

There are however quite a few ethnic French individuals from France that are definitely involved in upper tier organized crime; mostly with connections to the Corsicans and the Voyageurs. In Brussels in the 80's there were a couple of French brothers, originally from a town at the Belgian border, active in large scale narcotics trafficking, prostitution, forgery, art theft... One in particular was probably one of the biggest and most successful criminals in Belgium ever. Very well protected individual with far-reaching law enforcement and "high society" connections.