Suppose $\kappa\leq\mu$ are infinite cardinals. Let us agree to call a family $\mathcal{P}\subseteq[\mu]^{<\mu}$ a countably generating family for $[\mu]^\kappa$ if every member of $[\mu]^\kappa$ can be written as a union of countably many elements of $\mathcal{P}$. Note that we can extend this in the obvious way to structures of the form $[\mu]^{<\kappa}$ as well.

Theorem: If $\mu$ is a strong limit singular cardinal of uncountable cofinality, then the minimum cardinality of a countably generating family for $[\mu]^{{\rm cf}(\mu)}$ is the same as the minimum cardinality of a countably generating family for $[\mu]^{<\mu}$ (and hence, the same as the minimum cardinality of a countably generating family for $[\mu]^\kappa$ for any $\kappa$ with ${\rm cf}(\mu)\leq\kappa<\mu$).

The only proof I have relies on heavy machinery from pcf theory, but I do not believe this should be required, and I think that a more direct proof would shed light on some related questions. The following vague question asks for such a proof in a simple special case:

Question :Suppose $\mu$ is a strong limit singular cardinal of cofinality $\omega_1$, and let $\mathcal{P}$ be a countably generating family for $[\mu]^{\omega_1}$ (WLOG closed under subsets). Is there a reasonably constructive way to build a countably generating family for $[\mu]^{\omega_2}$ (of the same cardinality) from $\mathcal{P}$?

I don't have the language of Galois-Tukey connections available in this context to formulate this question more precisely, but I want to avoid tricks like "let $M$ be the Skolem hull of $\mathcal{P}$ and the other parameters, and then $M\cap [\mu]^{<\mu}$ works because the theorem is true".

Roughly speaking, the pcf-theoretic proof relies on Shelah's "cov vs. pp Theorem" to convert things into an equivalent question about pseudopowers, and then solves the associated pseudopower question using other results from his book *Cardinal Arithmetic*. Can we do better? I want to make sure I'm not just suffering from a blind spot and missing something easy.