Both are untrue, IMO, and in the desired steady state the car tax is in fact near zero, substituted by higher taxes on everything else. Even if that ends up making the city more expensive, the variation in utility is still at least positive if we model citizens' utility functions as negatively sloped on the pollution axis, and of course if we are assuming the central planning wants to comply with global warming goals.
I would even question if tradespeople would be against paying the car tax if it gets commuters out of the road, to be honest. I'd wager a plumber would be more than willing to pay even 100$ monthly if you worded it as "you get a fast pass to avoid all traffic and get everywhere as fast as the speed limit" and not "it's a tax on your car".