In my experience most of them do bring in someone for the stats aspect of it, but they're often not consulted on study design (when relevant), and their "expert

In my experience most of them do bring in someone for the stats aspect of it, but they're often not consulted on study design (when relevant), and their "experts" typically seem to be just "experts relative to the other authors" rather than actual stats experts. But that doesn't change the probable truth in your final statement.