Why does removing all copies of a letter preserve regularity?
Given an NFA or a DFA (like in your proof, if you want), we replace each arc with letter $a$ by $\varepsilon$. We obtain an NFA for $P(a,L)$. So, $P(a,L)$ must be regular.

In details, each trace in the newly constructed NFA can be transformed back to $a$-full instance of $L$ .