As an ex undertaker can I just say that during my working years we always gave our departed ones a shroud and a coffin as a shroud is the sometimes sheet the person is wrapped in. Today mourners prefer to used their loved one’s own clothes so we don’t refer to that/them as shrouds but I can assure you they are still used. We have always kept a supply of different styles in stock and pyjamas for the gentlemen and nightdresses for the ladies.
A coffin was always made of wood to carry the deceased in during the funeral, shroud around the body, coffin lined with silk panels but now mainly polyester in lovely colours. . Different style coffins today, wicker, cardboard etc definitely require a shroud which we would advise.
In past years most deceased were buried wearing shrouds as due to lack of finance it would be considered a terrible waste to use perfectly wearable clothing in the coffin, as someone would have dads suit, or mums best dress, common sense really. Bodies still come back to the funeral home from hospitals, nursing homes wearing a shroud or wrapped in a sheet.