Large spin system and Lipkin-Meshkov-Glick (LMG) model

I read about spin systems like LMG originally modelled in order to test the validity of approximation techniques in quantum many-body systems. But it’s also used to delicately understand the quantum criticality and the magnetic properties of atoms and molecules. The advantage of large spin system is that there is a classical limit and easy to analyse many phenomena. Classical limit is obtained for $Sgg 1$ where the commutator bracket of scaled spin operators tends towards zero
$$[s_x,s_y]=ifrac{hbar}{S}s_zrightarrow 0$$
Where $s_j=S_j/S$. But is it not mean that the quantum fluctuation is decreasing to zero, if it is true then how quantum phase transition occurs in these systems as quantum phase transition is mediated primarily by quantum fluctuation?