NOTE: @Ammarpad has pointed out that there is not a straightforward migration path right now for skins that need to support Cologne Blue, Monobook, Modern etc as these do not support SidebarBeforeOutput right now. One option is for extensions to support both SidebarBeforeOutput and SkinTemplateToolboxEnd for now, with the latter being removed later when skins have been updated and it's safe to do so. I will provide more clarity soon.
The BaseTemplateToolbox hook adds logic to the rendering layer.
It's used in a few places:
https://codesearch.wmflabs.org/search/?q=BaseTemplateToolbox&i=nope&files=&repos=
With T252917 the SidebarBeforeOutput hook will be usable instead.
Let's deprecate the hook and move usages to SidebarBeforeOutput.
= Acceptance criteria
[] All extensions in subtasks should use SidebarBeforeOutput (Vector) AND BaseTemplateToolbox (Modern, Monobook, Cologne Blue)
[] Soft deprecate BaseTemplate::getToolbox