The mole is defined to be exactly $\mathrm{mol}=6.02214076\cdot 10^{23}$ entities. Avogadro's constant is defined as $N_A=6.02214076\cdot 10^{23}\mathrm{mol}^{-1}$.

Then why is it not usual to write $$N_A=6.02214076\cdot 10^{23}\cdot (6.02214076\cdot 10^{23})^{-1}=1?$$

It doesn't make sense to me.