Clean way of building simple taxonomy browser of arbitrary depth
I figured out a way to do this with only the Views module. It's not perfect — it doesn't generate a nice URL and breadcrumb with the taxonomy lineage the way that my solution for a taxonomy of a single depth would do. 

Static Depth Browser

For a taxonomy of static depth (where all terms have the same depth of child terms), you would do the following:


Create a view of nodes
Add relationship to taxonomy term (Term)
Add relationship to taxonomy term parent, accessible via Term (Parent) 
Add relationship to taxonomy term parent, accessible via Parent (Parent Parent)
Repeat step 4 as many times as necessary for your taxonomy's depth.
Add contextual filters for each one and make sure they're in the proper order. Set to display a summary when filter is not in URL.
You'll probably want to override the breadcrumb and title for each filter.
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PROFIT


Variable Depth Browser

For a taxonomy of variable depth (where some term trees have deeper levels of nesting than others), it's a bit more complex:


Add a new top-level term (Foo) to your vocabulary and put all other terms under it. 
Create a view of terms with a page that displays term names
Add contextual filter: Taxonomy term: Parent term
When the default value is not in the URL, provide default value
Set default value to your top-level term name (Foo) 
Specify validation criteria


Validator: taxonomy term
Criteria: your vocabulary name
Filter value type: Term name converted to term ID

Click the "+Add" button at the top of the Views panel to add an Attachment to the view.
Add to this attachment only (override) Relationship: Taxonomy term: Content with term
Add to this attachment only (override) display: fields: node titles linked to nodes, and remove (from this attachment only!) display: fields: term name
Add to this attachment only (override) contextual filter: Taxonomy term: Name
When filter value is not available, provide default value: Raw value from URL: Path component: 2 (or more if your Page path has >1 component)
Attachment settings: Attach to: [name of the page display you just made]
???
PROFIT


Screenshots of Views' UI for reference: