After a bit of mucking around, it looks like onenote doesn't read the registry value for the quick access toolbar during start up. Oddly however, it does write to it when you change the setting while onenote is open.
I denied set permissions for my own user account on the registry key:
HKEY_CURRENT_USER\Software\Microsoft\Office\15.0\Common\Toolbars\OneNote\QuickAccessToolbarStyleFPV
This key is used to define what mode the ribbon and quick access toolbar in full page view (the view in question) are set to. The other DWORD 'QuickAccessToolbarStyle' is readduring start up and will determine how your ribbon is displayed
when in "normal" view.
Even after setting the ribbon to display in full page view in the registry and denying permission to change it, it just ignores it completely and opens with everything hidden anyway.... extremely annoying.
I reckon this would take about 20 minutes for a programmer to change and only slightly longer for QA to make sure it works.
I'm a very heavy user of onenote and use it professionally, for my study, and for my personal diary/daily planner. This is a real oversight and although a 'minor' problem on it's own, becomes a major burden when using onenote heavily.
Can this please be changed already!
I even went ahead and created a uservoice suggestions. Please upvote the issue here:
https://onenote.uservoice.com/forums/256328-onenote-users/suggestions/6802370-remember-ribbon-settings-when-in-full-page-view-in