Of course they don't. Multiple major crossings from Brooklyn, Queens, and the Bronx have no tolls into our out of Manhattan. Am I misreading you? Are you only talking about commuters from New Jersey?
The Brooklyn Bridge, Queensboro, Manhattan, GW (heading to NJ only), and Williamsburg bridges are the only bridges that do not currently have tolls. All cross the East River. With this plan, you will have to enter the congestion area to use all of those but the GW -- but that's already tolled heading into Manhattan. They even defined the "zone" to be sure to catch the Queensboro onramps. It's intentional.
As for NJ (which is every New Yorker's favorite scapegoat when imagining car commuters): every crossing of the Hudson south of Albany has a toll.
I guess you can count the tiny bridges over the Harlem river if you really want to be pedantic about it, but this is essentially a new tax on NYC residents, whether you own a car or not. Literally every truck entering Manhattan by a major artery will be tolled in one form or another -- in many cases, twice.
Yes, exactly.
Also: they already do. So consider that for a second.