Believe it or not
Have had great success in removing red wine stains by using Tide cold water "free and clear". Have soaked stained table linens overnight and next morning usually all traces of stains including red wine are gone. For the remains or stubborn spots then resort to bleaching. Mind you these were fresh stains. If things have been let go for awhile them YMMV.
My favourite go to stain remover for a host of stains is from Germany called "Hascherpur".
Got several tubes years ago and now find it has been discontinued. But am here to tell you the stuff shifts all sorts of color stains caused by wine, berries, tea, coffee, ink, etc.. Hascherpur is a reducing bleach; meaning unlike say sodium percarbonate or LCB it removes oxygen not adds. Those into dying would know reducing bleaches as "color removers".