Umm, no, not really:
A great deal of those costs fall on individuals rather than the government, bereavement, noise, congestion, air quality, essentially standard of living costs borne by the wider society.
In fact, if you raised that extra £20bn in taxes from motorists (increase fuel duty by 40p, probably more like 50-60p to account for reduced sales due to changing behaviour)
Many of those externalities identified are a result of having a transport system, not a result of having an ICE propelled car based one.
So even in theory we shouldn\’t be charging all of the costs of the externalities to the fuel.
It\’s entirely possible to build steam powered cars: with coal fired boilers n\’ll. There have even been cars powered by rotting chicken shit. Taxing petrol does not change the noise, congestion etc produced by those.