This sprawl inducement is also what makes it harder to build good efficient public transit. Investment in car oriented infrastructure degrades transit effectiveness and intrenches a city and its residents in a car oriented mode.
You're proposing to add to the cost of driving until that side of the equation exceeds the cost of urban housing. But if you don't address density restrictions or minimum parking requirements or urban property taxes, that is not going to end well.
Today it costs $3000/month for an apartment in the city vs. $1500/month for the same apartment in the suburbs and $400/month worth of driving cost. If you add $1100/month worth of driving fees to even it out but still have all the restrictions on creating new urban housing, next month the same apartment in the city will be $4000/month and 95% of the same people will still live in the suburbs. And employers will start to move out of the city because they can't afford to pay employees what it costs to work there anymore.
Whereas if you eliminate the impediments to creating new urban housing (or subsidize it if necessary), the price of the apartment in the city falls to $1700/month and we have the result we need without having a regressive tax on driving.
Essentially new development in greenfield or brownfield areas shields established urban residential areas from needing to be redeveloped.