Apparently, the default WordPress style files overwrite a custom TinyMCE skin. So what you need to do is deregister WordPress’s TinyMCE skin.
Since I needed the custom skin to only apply on the front-end on my website, I wrapped it inside an !is_admin()
conditional.
function remove_editor_buttons_style() {
// If not on wp-admin
if ( !is_admin() ) {
wp_deregister_style( 'editor-buttons' );
}
} add_action( 'wp_enqueue_scripts', 'remove_editor_buttons_style' );