Thank you for waiting, as I performed some tests on your website's clone.
The user form "Upgrade for VIP membership" on your website is different from a regular form because it is set to add a WooCommerce product to the checkout process as well.
For this particular form, you'll need to make the following changes to make this shadow date picker field work:
In the form's field editor, you'll update the existing shadow field code from:
I got this after I apply your code, the field is a text input field, I also tried to change the type to date and it unable to update the dob field data.
My issue is resolved now. Thank you!
Last week we migrate the site to another hosting server. The date choosing option shows the date data much faster.