There's no always-applicably configuration. Builds triggered by SCM changed or by timer (cron) don't have a "User who triggered the build" associated with them, so they will run as SYSTEM unless overridden in the project config.
The priority order of these authorizations is from top to bottom and the first applicable will be used. Therefore it makes no sense to place the default "User who triggered the build" higher than the project overridable "user who triggered the build", and projects configured to run as anonymous will only do so if no "User who triggered the build" can be determined.