Note: I am posting an answer to my own question with hopes of it being useful for developers trying to solve the same problem that my initial query proposes.
As it turns out, I was viewing this with the wrong lens. The solution was not to add my entire AngularJS app and use that while trying to disguise the WordPress front-end and themes. The solution is actually to create a WordPress theme that uses my AngularJS app as a static starting page, and build the app thereon.
I was inspired by this article: https://1fix.io/angularjs-wp-rest-api/
I can just build an AngularJS app and base it on the index.php file of a new theme. It was as easy as that. I guess I just couldn’t see the forest for the trees.