No, that doesn't follow: entropy can decrease provided that it is exothermic enough. For a spontaneous reaction to occur, the relevant quantity isn't the entro

No, that doesn't follow: entropy can decrease provided that it is exothermic enough. For a spontaneous reaction to occur, the relevant quantity isn't the entropy, but the Gibbs free energy. So reactions that reduce entropy (because "things crystalize out") can take place if the enthalpy increases enough to compensate.