Taxonomy view : add token term_name in url
This does what expected.

In the content pane view, Contextual filters : Content: Has taxonomy term ID (with depth), use the filter value type "Term name converted to term ID"



In the custom page, in basics, set the path almost as in the question destinations/[term:name]/foo (I abandoned the [term:parents] part, no way).

And in arguments, choose "argument type:term name".