Feel free to skip to the question below; the following is just context and discussion:

An interesting, but seemingly less used result in the theory of nearby cycles of constructible sheaves is (roughly) the following.

Let $f: X \to D$ be a proper, stratified analytic map between a Whitney stratified complex analytic space $X$ and the open unit complex disk $D$. Suppose that the sheaf $\mathcal{F}$ is constructible with respect to the stratification on $X$. Then the nearby cycles $\psi_f(\mathcal{F})$ can be expressed as the pushforward $r_*(\mathcal{F}|_{f^{-1}(t)})$ along a proper 'retraction' map $r: f^{-1}(t) \to f^{-1}(0)$ (this map will depend on the stratification of $X$!), where $|t|$ is sufficiently small.

For the exact statement, there are scattered sources, but the standard one seems to be section 6.13 of Stratified Morse Theory by Goresky and MacPherson (also see remark 5.5.1 in these notes). Though I haven't been able to pin down a complete proof of this statement in the literature, I'm happy to accept it as true for this question.

Knowing that you can express nearby cycles as a proper pushforward in context seems like it would be useful. For instance, consider the situation in which we might take iterated nearby cycles: $f_1 \times f_2: X \to D^2$. If we suppose that $f_1$ and $\mathcal{F}$ satisfy the hypotheses above, then $\psi_{f_2|_{f^{-1}(0)}}\psi_{f_1}(\mathcal{F}) \simeq \psi_{f_2|_{f_1^{-1}(0)}}(r_*(\mathcal{F}|_{f^{-1}(t)})) \simeq (r|_{(f_2 \circ r)^{-1}(0)})_*\psi_{f_2 \circ r}(\mathcal{F}|_{f_1^{-1}(t)})$, which seems related to the question of whether iterated nearby cycles commute.

Question: I haven't seen the very geometric, Goresky-MacPherson type approach to nearby cycles in modern literature. Why? How can I see that something like the discussion above is subsumed (in the constructible, topological setting, knowing it won't make sense in other contexts) by the algebraic theory in which nearby and vanishing cycles are dealt with though sheaf-theoretic identities and arguments using various diagrams of spaces with certain properties (e.g. complementary open and closed subspaces)? Is there any place in the modern theory for taking tubular neighborhoods and playing with these spaces like Play-Doh the way people did in the '70s-'80s?