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.