Drupal 9: Adding a form on a page with AJAX - drupal

Question

I've got a controller that is generating some markup. In the markup there is a link that dynamically adds some content to the page using jQuery ajax.
The content that's added to the page comes from another controller. In that content there's a Drupal form. It renders correctly on the page, but it doesn't submit properly (although it works well if I call it directly - it only fails to submit if it's inserted through AJAX).
So, in summary, controller A renders some markup like this:
<a id="ajax-link">Click here to display the form</a>
<div id="form"></div> <!-- form will be displayed here through AJAX -->
In the javascript, I've got:
$("#ajax-link").on("click", function() {
$.ajax({
url: "/controller-B/get-form",
type: "POST",
dataType: "JSON",
success: function(data) {
$("#form").html(data.output);
}
});
Controller B renders the form like this:
public function get_form() {
$form = \Drupal::formBuilder()->getForm('Drupal\my_module\Form\MyForm');
$output = \Drupal::service('renderer')->render($form);
$data = new \StdClass();
$data->output = $output;
return new JsonResponse($data);
}
So I click on the link, it calls the script, which calls controller B, which generates the form, and the script adds the markup for the form on the page.
It works well, but the form doesn't submit properly.
Looking at the markup that's generated after the form is inserted on the page, it looks like it's the action attribute that's incorrect. It generates something like this:
<form action="form_action_p_pvdeGsVG5zNF_XLGPTvYSKCf43t8qZYSwcfZl2uzM" method="post" id="comment-form" accept-charset="UTF-8" class="comment-form" data-drupal-selector="comment-form">
Any tip would be greatly appreciated!