Much clearer now, thanks.
The TinyMCE Advanced plugin only extends the default configuration of TinyMCE, it doesn't change which editor is default.
In core that setting is stored in a cookie every time you click on the "Visual" or "Text" button and later saved in the DB. A plugin or a theme can change/disable this but don't see why. You can see what's there by pasting this in the browser console:
getUserSetting('editor') and set it with
To check on the PHP side, paste this somewhere in edit-form-advanced.php:
<?php echo get_user_setting('editor'); ?>.
And you can set that by adding:
in a function running on 'admin_init'.