Force the “Choose from the most used tags” meta box section to always be expanded

You can do this using jQuery.

First of all, you need to add your script into your functions.php

function my_theme_scripts() {
    wp_enqueue_script( 'expand-tags', get_site_url() . '/js/expand-tags.js', array( 'jquery' ), '1.0.0', true );
}

add_action( 'admin_enqueue_scripts', 'my_theme_scripts' );

Then, in your file (expand-tags.js) which is located on, for example, public_html/mysite/js/, you will have to have something like this:

$(document).ready(function() {
   $("#link-post_tag").trigger("click");
});

link-post_tag is the ID of the Choose from the most used tags button, so when the page is loaded, a button click will be triggered.