$$f(x)=\dfrac{1}{\sqrt{2\pi \sigma^2}} e^{-\dfrac{(x-\mu)^2}{2 \sigma^2}}$$
The normal distribution has parameters $\mu\in\mathbb{R}$ and $\sigma\in \mathbb{R}_+$ as above, and regardless of their values, the function integrates to $1$ over $\mathbb{R}$. However, if instead of the above function we write $$f(x)=\dfrac{1}{\sqrt{a\pi \sigma^2}} e^{-\dfrac{(x-\mu)^2}{a \sigma^2}}$$
with $2$ replaced by a new parameter $a$, it still integrates to one, and is a valid distribution. Why is this never used? If you add a new parameter, doesn't it make it easier to find parameters to "fit" the data better rather than with just $2$ parameters, since this new parameter allows for a more flexible distribution?
If we introduce this new parameter, firstly it is redundant (it can be absorbed in $\sigma^2$) and secondly it means that $\sigma^2$ is no longer the variance of the distribution in general. It is the variance exactly when $a=2$, which is why the constant is set to this value.
Edit: I forgot to make it explicit that no, this does not give extra "freedom" of the function, exactly because it can be absorbed in the $\sigma^2$ parameter.