You won`t need SkyClub membership card if you are flying international as your PM status has access to SkyClub lounges when flying international.
But if flying solely US domestic Delta flights or other airlines flights then you need to present SkyClub membership card for lounge access, especially partner airlines lounge they will definitely ask you to show SkyClub membership card as they recognize SkyClub membership card is the only lounge membership card and PM GM card is just Elite status frequent flyer card which only have access to Delta and affiliate lounge when flying with them.
That is the reason why Delta mails SkyClub membership card separately apart from GM PM card just in case passenger flys with other airlines or US domestic solely.