Found the answer :
wp_set_current_user(1); // IMPORTANT FOR CRONJOB
$request = new WP_REST_Request( 'PUT', '/wp/v2/pages/'.$pageID );
$request->set_query_params( [ 'content' => $new_content ] );
$rest_response = rest_do_request($request);
wp_set_current_user(0);
In case someone needs it ;)
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