Hi,
I think I found what is happening. It is not the ‘hidden’ attribute, but the script that open the dialog window causing the issue.
When clicking the ‘Show dialog’ button, the script does changes to the search settings form, adding a ‘disabled’ attribute to the post type option, so basically removing the post type options, see: https://youtu.be/39LbY1NhfRM
That is not related to the search script unfortunately. Try looking for the dialog script source, to prevent this behavior. If you check the video closely, many other elements on the page are added some attributes (data-cached-aria-hidden and aria-hidden), that I don’t think it should happen.