Text field will not save after moving it from commerce billing profile to account fieldset in checkout
Unfortunately, you can't move this and have it continue working as normal. Entity form submission depends on a specific "tree" hierarchy in the submitted field values, and moving it like this disrupts that. Thus when the customer profile information is saving, your custom textfield is just invisible.

If you really need to move this, then you should add a custom submit handler that populates the customer profile as before.