I've done a few more tests and I can say that the fields are fine, the problem is essentially this user post type.
Perhaps there is a hidden naming conflict. Can you remake the post type under a different name with a different slug.
In my test I created a CPT known as User Profile and this worked. All I did was to switch your user field group to this new post type and set the form to create posts for this post type and it worked.
"In my test I created a CPT known as User Profile and this worked. All I did was to switch your user field group to this new post type and set the form to create posts for this post type and it worked"
What I meant was that I created a new Post Type called User Profile. This would mean that you will get a new post type with the slug user-profile. Do not use the original User post type that you've created.
The next thing that I did was to go to the custom field group and instead of having it assigned to the User post type I reassign it to the User Profile post type. Then I went to your form and made the change to the User Profile as well.
Once I did this the form fields begin to show up fine.