How can I create a view page that takes taxonomy machine name and has depth?
You can use the contextual filter Content: Has taxonomy term ID (with depth) without adding a Relationship.  Use this settings for the contextual filter:

When the filter value is NOT available: Provide default value 
Type: Taxonomy Term ID from URL
Make sure Load default filter from term page is checked.

Please note: When you visit a pathauto destination path, in this case with a source of /taxonomy/term/% pointing at /vocabulary_name/term_name, drupal passes the source path as a query variable, and your view still uses the source path to gather information.  This means your path in the view should not change, it should stay as taxonomy/term/%