If the closing is a “mail away” we usually need the package ready several days before closing. Does the three-day time period also apply to the closing documents from the lender or is it just the Closing Disclosure (CD)?

The CD is the disclosure document required to be delivered prior to closing, not the entire closing package. Therefore you are going to have to do what you have always done: stay in close communication with the lender in order to get everything you need when you need it.