I'm a bit confused about the Hilbert-style axiomatization of first-order logic. More precisely, I am a bit confused about completeness w.r.t. to Hilbert-calculi. A complete Hilbert-style calculus I am familiar with uses e.g. the quantifier rule
$$\frac{\phi\rightarrow\psi}{\phi\rightarrow\forall x\psi}$$
for $x$ not free in $\phi$. Now, if we allow for derivations using assumptions, we get that
$$\{\phi(x)\}\vdash\forall x\phi(x)$$
but clearly $\{\phi(x)\}\not\models\forall x\phi(x)$ where $\models$ is classical semantical entailment. That means that the above rule is not sound for derivations with assumptions using open formulae.
Now, in some literature I have read, they avoid this by only showing soundness and completeness w.r.t. to sets of closed formulae, which I dislike as I was taught first-order logic using sequent calculi, where this restriction is not necessary as we can include requirements for the assumptions of the proof in the rule, e.g. by considering the rule
$$\frac{\Gamma\Rightarrow\phi\rightarrow\psi}{\Gamma\Rightarrow\phi\rightarrow\forall x\psi}$$
for $x$ not free in $\Gamma$ and $\phi$.
This leads me to the question:
Is the restriction to closed formulae (in assumptions) an inherent limitation of the Hilbert-style approach to first-order proof theory or can this be circumvented?
