This is another good question and I presume it's being asked for situations in which you are signing on a day that is NOT the date of closing. Well, I have my opinion and I will share it, however, realize that the mortgage lender will really make the final decision.
Let's start by saying we really have two dates. We have the actual date of signature and we have the effective date of the document. In a perfect world mortgage lenders and recorders of deeds would understand the difference.
In that case, we could type on the first page of a Note, let's say, the word "effective" next to the date and then that date could be the date of closing. We could also type the word "effective" date on the first page of the Mortgage and leave the notary section as the actual date the document was signed. Everybody would understand that though folks signed the documents on an earlier date, the effective date was later.
Run this method by the mortgage lender. Hopefully, they will understand.