Word 2013’s Quick Parts > Document Property menu is greyed out

Record a macro for these actions and assign it to a single keyboard shortcut. This is the exact usecase macros are made for.

Turns out the date field does update automatically, but only after closing and re-opening the document.

Note: The date may not update in your document unless you close, then re-open the file or right-click the field and select “Update Field“.

Now the date in your document will be inserted into the document and will update automatically based on the selection you made whenever you open the [Word] document.

_{Technipages - Word 2016 & 2013: Insert Date that Updates Itself Automatically}