‘all’ hook and add_action issue (class based plugin)

First, template_redirect is a front end hook. It never fires on your backend save. You need to choose a backend hook such as admin_init to hook your function.

Secondly, based on your pastebin from you other question, your code has the checked attribute hard-coded for both of your radio boxes, thus causing it to default to the last one all the time. You are going to want to edit your form like so:

<input type="radio" <?php checked($this->data, 1); ?> value="1" name="selection" >

Third, you need to make sure that both of your radio buttons have a value attribute.

Fourth, you need to update the option when you save so…

public function on_admin_form_submit() {
    if ( isset( $_POST['selection'] ) ) {
      if ($_POST['selection'] == 1){
        update_option( 'where_to_log_to', 1 );
      } else {
        update_option( 'where_to_log_to', 2 );
      }
    }
    $this->data = get_option( 'where_to_log_to', 1 );
}

Notice that I explicitly saved the values 1 and 2.

I think that’s got it.