Working in the first order language of set theory.
By $R$-bounded quantifiers its meant those of the form $\forall x \ R \ a \, ( \cdots) $ , or $ \exists x \ R \ a \, ( \cdots) $, and these are defined as $\forall x \, ( x \ R \ a \to \cdots)$ and $\exists x \, (x \ R \ a \land \cdots) $ respectively. Here $R$ is a relation symbol.
A quantifier is defined here to be open if it is neither $\in$-bounded nor $\subseteq$-bounded.
Let $\varphi^{\mathcal V|}$ be a formula obtained from $\varphi$ by merely $\in$-bounding some open quantifiers in $\varphi$ by the symbol "$\mathcal V$" in a manner that is closed anteriorly; that is, if an open quantifier is bounded by $\mathcal V$ then all open quantifiers to the left of it must also be bounded by $\mathcal V$.
Anterior Bounded Reflection: if $\varphi$ is a formula that doesn't use the symbol "$\mathcal V$", then: $$ \forall \vec{a} \, (\varphi \to \exists \mathcal V : \varphi^{\mathcal V|})$$
This by itself can prove: Pairing, Union, Powerset, Infinity, Replacement, Collection, Reflection on transitive sets, and Reflection on super-transitive sets.
So, this principle together with Extensionality and Separation can prove all axioms of $\sf ZF-Reg.$ in a straightforward manner! And just coming from basic ground level set theory language, so it doesn't require defining transitivity nor super-transitivity nor stages of the cumulative hierarchy which are not ground level language tools.
Seeing how this can be expressed easily and in such a parsimonious manner, then is it eligible to be placed among the main basic principles of set theory, i.e. at axiomatic or near axiomatic level?
My 2c, as for one thing I am certainly not an expert of set theory:
On the "philosophical" side, I would not think there is a definite answer there: what one adopts as a "principle" is partly in the name of economy and easy proofs, partly in the name of understandability and cogency, in other words, for a clear informal reading. For example, take the "Sheffer stroke", which is alone sufficient to bootstrap boolean arithmetic, but makes understanding more difficult, e.g. just think how "not" is expressed in those terms. Indeed, for "principled" definitions, I would think the semantic side is eventually more important than the syntactic one.
On the technical side, there is indeed, relevant to set theory, a "reflection principle" (thank you!). And I would think the axiom you present might indeed fall under that notion, although not strictly: Wikipedia says <<The name "reflection principle" comes from the fact that properties of the universe of all sets are "reflected" down to a smaller set>>, what you are doing is more like being free to assume universes ("(implicit) universes of discourse" I'd be inclined to dub these) in order to close any formula, and it's in the opposite direction, bottom-up and without assuming any properties about those universes (AIUI). And that such a seemingly innocuous axiom lets us derive most axioms of set theory I find pretty amazing... to the point that I would definitely reply with a yes to your question. (Indeed, also interesting would be to study what it does not recover of the usual axioms.)