We have $\textbf{r}(s,t)$ as the parametrization of a surface. Then the element of surface area is defined as $$dA = \left|\frac{\partial \textbf{r}}{\partial s}\times\frac{\partial \textbf{r}}{\partial t}\right|dsdt$$ where $|\cdot |$ represents the modulus.
Also, we know that the vector element of area $d \textbf{S} = \hat{\textbf{n}}dA$.
Hence we can write $$d \textbf{S} = \hat{\textbf{n}}dA=\hat{\textbf{n}}\left|\frac{\partial \textbf{r}}{\partial s}\times\frac{\partial \textbf{r}}{\partial t}\right|dsdt$$
but now recall that $\hat{\textbf{n}} = \left(\frac{\partial \textbf{r}}{\partial s}\times\frac{\partial \textbf{r}}{\partial t}\right)$ by definition! Therefore $|\hat{\textbf{n}} | = \left|\frac{\partial \textbf{r}}{\partial s}\times\frac{\partial \textbf{r}}{\partial t}\right|=1$ which tells us that $dA = dsdt$ and $d\textbf{S} = \hat{\textbf{n}} dsdt$.
Then why don't we use these shorter notations? Furthermore, is there a flaw in my reasoning?