Part of the issue here is that if you own a car, it’s often cheaper to take the car than public transport, because most of the car expenses are paid independent of the immediate usage.
Car value deprecation, taxes, maintainance, all of that cost you money no matter whether you drive into town today or use some other means of transport.
I think it would be much better to put all taxes onto the fuel price. If you pay €5 for a litre of fuel, instead of the ~€1.5/l that we are currently paying, it would make more sense to take public transport some times.