You don't need to trade a physically settled contract for that though! Just hedge on a cash basis or hedge directly on the item you do need delivery on.
Seriously, do people enter into contracts like this in other parts of their life?
Cash settled contracts add a layer of indirection: we have to agree on a method to determine the price of physical oil on our preferred date. There is a popular cash-settled future for WTI crude. It trades on ICE and uses the settlement price of the CME future as its reference price, so exactly the same issue arises there.
As for trading exactly the item you will need delivery on, that may be hard to find. Standardising on a contract that's "close enough" allows crude oil producers to trade with airlines, bus companies to trade with refineries, etc, even if they all care about different products.
Seriously, do people enter into contracts like this in other parts of their life?