I'm asking-and-answering this question in case others run into the same issue.
I'm running Enterprise Search, Elasticsearch, and Kibana 7.14.0. I created an Elasticsearch native user using the Elasticsearch API (I did not use the Users and Roles UI within Enterprise Search to create the user -- that works fine). I logged into the Enterprise Search standalone UI using this native user. I clicked the "New interface, new possibilities" link to launch the new beta interface in Kibana.
Kibana showed me the error, "You do not have permission to access the requested page."
This error appears because the Elasticsearch user I created did not explicitly have access to Kibana.
To resolve the issue, I granted the user the enterprise-search-user Elasticsearch role, which gives the necessary access to Kibana and the Enterprise Search Kibana plugin.