You will probably find substitutes will Spring Up for the uses of 'little oil' like natural gas which has displaced a lot of the feedstock for plastics in the US and products like Lithium grease and silicon are already quite capable lubricants. Bio-oil from waste and other alternatives can be employed for Asphalt, Nylon, Polyester and Bio-resins used for fabrication etc etc.
It really will boil down to innovation and beating the cost curve of oil based products to kill off 'little oil', as long as we don't revert back to boiling down Whale Blubber* to keep the wheels of civilization turning ........ (*mankind slaughtered 20 million whales in the 20th century)