See other parts of the same page: access= is used to set the default access for the way. For example, https://wiki.openstreetmap.org/wiki/Key:access#Transport_mode_restrictions

I suspect something else is going on, because OpenRouteService seems happy to use https://www.openstreetmap.org/way/26487015 for pedestrian routing, for example, and it has access=private foot=designated. See https://maps.openrouteservice.org/#/directions/Keldwyth%20Drive,Windermere,England,United%20Kingdom/Wynlass%20Park,Windermere,England,United%20Kingdom/data/55,130,32,198,15,97,4,224,38,9,96,59,2,24,5,192,166,6,113,0,184,64,90,0,152,3,160,19,128,70,0,89,200,13,134,202,5,96,1,153,167,40,3,157,129,216,1,160,114,226,1,153,73,55,174,74,151,114,21,42,8,13,196,76,185,38,229,6,20,164,221,149,66,76,70,11,224,48,123,26,237,40,142,227,43,160,193,32,120,128,128,1,213,60,8,137,176,229,0,11,202,0,91,92,229,120,131,219,64,64,1,155,192,0,219,162,224,128,133,66,163,224,0,91,192,3,89,32,3,152,218,7,67,160,135,163,102,33,129,69,227,102,67,123,123,162,34,195,162,192,128,2,250,213,0,0