Am I using the right hook for removing quicktags on the admin TinyMCE?

After checking out the code, the best way to do this would be to use the wp_editor_settings filter in /wp-includes/class-wp-editor.php. When you call wp_editor() it internally makes a call to _WP_Editors::editor($content, $editor_id, $settings);. This function first passes the $settings array through parse_settings() which uses that filter.

add_filter( 'wp_editor_settings', 'remove_editor_quicktags', 10, 2 );
function remove_editor_quicktags( $settings, $id ){
    // $id will be 'content' in your example
    // use it in an if or make it gone for everything...

    // use $pagenow to determine if you are on the edit comments page.
    global $pagenow; 
    if ( $pagenow === 'comment.php' ){
        $settings['quicktags'] = false;
    }
    return $settings;
}

Note – I just realized this filter is new as of WordPress 4.0, so you will need it or newer to take advantage. This also affects all instances of TinyMCE on the admin.