Maybe you could do it with a plugin if someone could make you one for you
Other than yes to add a token showing the current date you probbaly would need to change the source code
But as an alternative "%Order.CreatedOn%" - Is the Gift card valid from when the order is placed ? granted the format is probably longer than what you want.
For Code (plugin) change
You need to edit
AddStoreTokens(IList<Token> tokens, Store store, EmailAccount emailAccount)
tokens.Add(new Token("Store.DateToday", DateTime.UtcNow.ToShortDateString()));
Then use token %Store.DateToday% in the message template