Update:
It seems like the override settings that I need in my config.php should be:
$default_advanced_search_mode="2";
# Document only
This works for advanced search, but does not display the type-specific advanced search functionality until unchecked and rechecked, which is only the more unintuitive from a UX perspective.
$default_res_types="2";
# Document only
For basic search in the right nav. This does not seem to work.
Ideally, we would like to restore the previous functionality from 9.8, wherein dropdowns and type-specific search fields are accessible regardless of what is checked in either search tool.