Optimal insertion times in insertion-only data structures beyond Bentley-Saxe
Think about your question in reverse: suppose you have a dynamic data structure for some problem — does that imply that you can solve it statically, faster by a log? Why should it? And in fact it is not true.

Consider range counting in one-dimensional intervals, in a comparison model of computation. That is, the data is a set $S$ of numbers, the query is an interval $[\ell,r]$, and the answer to a query is the size of $S\cap[\ell,r]$. Statically, you can solve it in $O(\log n)$ query time by storing $S$ as a sorted array and using binary search, in preprocessing time $P(n)=\Theta(n\log n)$. Dynamically, you can still solve it in $O(\log n)$ query time by using balanced binary search trees, with insertion time $O(\log n)=O(P(n)/n)$.