This is a bad approach. At least if its only that.
In Manhatten this works because there is already a decent public transport, already a culture of waking and an established culture of biking.
You can just force people into better cities with punishing taxes. You actually have make the roads safer, provide alternatives and so on. And this is easier said then done, almost all cities in the US have zoning codes and other laws that make it completely impossible to build decent urban infrastructure. And the traffic standards are literally 100% backwards to providing safety.
In fact, because the traffic standards are so bad that less cars actually kill more people. This is because a lot of traffic slows down vehicle speeds on avg.
So basically, if all you are doing is forcing less people to drive, without doing anything else, you are just gone make the roads unsafer, and not improve the city or the lives of most people.
You can tax to such levels that you have both the funding for and the road space for a legitimately great bus or tram system.
This is unfortunately a problem where, as you say, many issues interact to make it difficult to solve piecemeal in most places. But that doesn't mean we can or should just allow it to fester. I agree zoning and physical infrastructure need to improve in tandem in most places.
Pretty trivial to discount/exempt people as is done in NYC.
An even simpler starting point (which we should actually do for all road-related fees like tickets IMO) is to set fees by the KBB value of the vehicle in question. Let people contest them in court if they want.
