How do I bulk update organization subtypes using a profile?
This problem sounded vaguely familiar, and I just found out why.

This is logged on the (old) CiviCRM issue tracker as CRM-17451.  The answer was that someone would have to write this functionality or work with the core team to do so.  However, I also see that I found a workaround back in 2016.  From the page:


  In non-technical terms: Profiles secretly have a contact type they're
  associated with. You can see this if you try to add an
  organization-specific field to a profile, then an individual-specific
  field. You'll get the error "Cannot add or update profile field type
  Individual with combination of Household or Organization". That's
  because Civi has decided this is a profile associated with
  Organizations. However, when you don't have any contact type-specific
  fields on a profile, your profile defaults to a contact type of
  "Individual" for many purposes. So individual contact subtypes are
  loaded.
  
  The workaround: Add an organization-specific field (like "Organization
  Name") to your profile, and this problem goes away.