How to remove admin footer text from inside child theme?

As suggested in the comments, these kind of modifications would be better served with a custom plugin, as they are not theme dependent. Here’s an example:

<?php
/**
 * Plugin Name: Remove Footer Text
 * Description: Remove admin footer text and update footer text through filters.
 * Version:     0.0.1
 * Author:      Name
 * Author URI:  http://example.tld
 */

add_filter( 'admin_footer_text', '__return_empty_string', 11 ); 
add_filter( 'update_footer',     '__return_empty_string', 11 );

Setup: Create the /wp-content/plugins/wpse-remove-footer-text/plugin.php and activate the plugin from the backend as usual.

If other plugins or the themes are adjusting the footer text, then we might need to increase the priority value.