It is customary to define the class of partial recursive functions by taking the set of primitive recursive functions $PR$ and taking closure over unbound search operation.

Do we need the "whole" set of primitive recursive functions as a base to reach the class of partial recursive functions with the closure? It seems reasonable to assume that unbound search could be used in the place of primitive recursion to reach the class $PR$ from a smaller base class.

For example Grzegorczyk proves that

Every computable function can be presented in the form $f(u)=A(ix[B(u,x)=0])$, where $A$ and $B$ are functions of the class $\mathcal{E}_0$.

Here $ix[B(u,x)=0]$ is the unique $x$ such that $B(u,x)=0$ and $\mathcal{E}_0$ is the lowest set in Grzegorczyk-hierarchy. I don't, however, see how to change the "unique $x$ such.." to "smallest $x$ such..".