First is fuel versus non-fuel usage of oil. Oil for non-fuel purposes can be drawn indefinitely with little climate impact, provided oil for fuel usage is phased out.
Second, on fuel usage you are correct there is currently no commercially-viable way to make it carbon-neutral. There may be in the future, and companies like Exxon are understandably working towards that. It might never happen, but that is not a reason to engage in sensationalism.