From the $so(3)$ Lie algebra $$ [\hat{s}^a_j,\hat{s}^b_k]~=~i\hbar\delta_{jk}\epsilon^{abc}\hat{s}^c_k ,$$$$ [\hat{s}^a_j,\hat{s}^b_k]~=~i\hbar\delta_{jk}\sum_{c=1}^3\epsilon^{abc}\hat{s}^c_k ,\tag{1}$$ it is clearfollows that $$\hat{\bf S} ~=~\sum_j \hat{\bf s}_j$$$$\hat{\bf S} ~=~\sum_j \hat{\bf s}_j\tag{2}$$ generates rotations.
Normalize $$\hat{\bf s}_j \leadsto\frac{\hat{\bf s}_j}{\hbar}$$$$\hat{\bf s}_j \leadsto\frac{\hat{\bf s}_j}{\hbar}\tag{3}$$ to get a Lie algebra (1) that survives the classical limit $\hbar\to 0$.
It is overkill to use Noether's theorem. It is just Heisenberg's EOM $$\frac{d\hat{\bf S}}{dt}~=~\frac{1}{i\hbar}[\hat{\bf S},\hat{H}]~=~\hat{0}.$$$$\frac{d\hat{\bf S}}{dt}~=~\frac{1}{i\hbar}[\hat{\bf S},\hat{H}]~=~\hat{0}.\tag{4}$$
