That's only true if driving is the only way to get to work. If people can get to work on public transit, and it costs less than paying the congestion charge, they will use public transit.
In other words, the congestion charge is a constraint on traffic volume, just as road supply is, because it changes the equilibrium of supply and demand.
It isn't like this is a hypothetical experiment that hasn't been tried. It is a fact that people take public transit when it is easier or cheaper than driving. It has happened in cities around the world.
Choice where I live: (1) Live in inner city with high taxes, high crime, poor schools, and nonfunctional govt, but have access to mass transit that doesn't go anywhere useful and takes an hour to get there. (2) Live in the burbs with lower taxes, good schools, low crime, and (sorta) functional govt but no mass transit. At least with choice #2 I can easily get wherever I typically need to within 30 minutes in my car.
Expanding mass transit isn't going to fix the reasons why people aren't interested in living in the cities hereabouts. It will just add to the subsidies already flowing in to prop them up.
In other words, the congestion charge is a constraint on traffic volume, just as road supply is, because it changes the equilibrium of supply and demand.
It isn't like this is a hypothetical experiment that hasn't been tried. It is a fact that people take public transit when it is easier or cheaper than driving. It has happened in cities around the world.