Insert the Date Picker CC directly from the Developer tab.



Do not copy the ones that are currently there. Those are mapped to the Publish Date Document Property.

If the date picker has been inserted correctly, even if the "Publish Date" content control has been used, when a new document is created the date picker will default to today's date.

The only circumstance that will result in the calendar showing a date in the past is if the content control in the template is displaying a date instead of the prompt text. For a standard date picker the default prompt text is "Click or tap to enter a date", whereas "[Publish Date]" is displayed if that control has been used.

The Publish Date Document Property Content Control is a mapped Content Control.

Yes, it will reset when a new document is created.
However, if it is used in different places in the document for different dates, each instance will change to meet whatever date is put into any of them.
Repeating Data Using Document Property Content Controls and Other Mapped Content Controls