Callback is not called in add_settings_field() when passed as part of an array, but recognises that it’s there. It’s passed Class to Class using OOP

From what I could see, the problem is not with the field callback, but it’s the section ID whereby you’re setting it to default — see $field["section"] = 'default' below:

// In the foreach in AddSettings.php
add_settings_field( $field["id"], $field["title"], isset( $field["callback"] ) ? $field["callback"] : "", $field["page"], $field["section"] = 'default', $field["args"] = array() );

So you’d only to correct that (typo?) and the fields would show up as expected, or that the field callback would run properly. And you should also correct the $field["args"] = array()..