Yes, you are allowed to rent a French car regardless of your flight.
To reach the French sector when you arrive: follow the same route as the other passengers, collect your baggage and then, once you are in the Arrivals hall, after going through customs, go up one floor to the Check-in level. Follow the “destination France" signs to reach the French sector (to the right of the terminal near the Air France ticket counters).
Warning: Following security measures imposed by the Swiss and French authorities, Genève Aéroport informs you that ONLY passengers in possession of an air ticket valid for day D +/-1 are authorised to pass between the international sector and the France sector.
Therefore, car rental agencies’ clients that do not meet this condition must imperatively move from one sector to another by road (car, taxi, etc.), by passing through the Ferney-Voltaire customs. Pedestrians and cyclists are not allowed to use this road.
Genève Aéroport thanks you for your understanding.