When you open the template itself (as oppose to creating a new document
based on that template), does a December date show as a placeholder? If it
does, then click the Developer tab and click the Design Mode tool so that
you're in design mode (this is not essential, but it makes it easier to
avoid deleting the whole content control rather than just the placeholder
date). Delete the placeholder date from between the two content control
symbols. Then click away from the date picker. It should now display as
"Click here to enter a date".
Turn Design Mode off, and save the template. When you create a new document
based on the template, the date picker should now display "Click here to
enter a date", and it should default to showing the current date.