Field with '#type' = 'hidden' in hook form alter doesn't show #id in the source and it's not possible to show it with jQuery
You've converted the element to a hidden input type.

Hidden inputs cannot be rendered, as the name suggests (they're hidden by their nature). So trying to use javascript to "show" them will never have any effect - there's nothing to show.

Add that to the fact that a changed hidden input element will invoke a validation failure for a tainted form, and it should be easy to see why the Drupal devs didn't bother to add an HTML ID to the element. Trying to access it with CSS/JS makes very little sense most of the time.

If you just meant to hide the original input, not convert it to a different input type, just use CSS to hide/show it as normal.