No. We don't need every primitive recursive function.
In fact every recursive function is the composition of two $\Delta_0$ functions with the minimization operator. Since every $\Delta_0$ function is primitive recursive, but not every primitive recursive is $\Delta_0$ this shows that you don't really need all the primitive recursive functions after all.
This is because we can write the graph of every recursive function as a $\Sigma_1$ set, i.e. $\exists z\varphi(\vec x,y,z)$ where $\varphi$ is a $\Delta_0$ formula. We can define from $\varphi$ two functions $G$ and $H$ as follows:
- $G(\vec x,w)=\min_{y < w}\exists z < w\varphi(\vec x,y,z)$ - this function returns the least $y$ such that $\varphi(\vec x,y,z)$ holds, if such $z$ exists.
- $H(\vec x,w)=\exists y < w\exists z < w\varphi(\vec x,y,z)$ - this function returns $0$ if there is such $y$ and $z$ below $w$, and $1$ otherwise.
Now $f(\vec x)=G(\vec x,\mu w H(\vec x,w))$, and one can verify that even if $f$ is a partial function the result is the same (i.e. it does not extend $f$).