If you put high congestion tolls on the freeways, people will just move their commutes to the smaller arterial roads that go between cities and jam them up. If you put congestion tolls on those, people will commute through residential areas. If you put congestion tolls on every street in the 500 square mile surface area of Los Angeles, you'll be voted out of office.
If you widen the roads or add lanes, making commutes less miserable, people will move farther from work where it's cheaper and maintain their previous misery level. In other words, traffic volume (demand) will always rise to meet the available amount of road (supply).
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.
If you widen the roads or add lanes, making commutes less miserable, people will move farther from work where it's cheaper and maintain their previous misery level. In other words, traffic volume (demand) will always rise to meet the available amount of road (supply).