Changes since congestion pricing started in New York

I wonder if this will eventually lead to increased density and if that then leads to congested bike lanes. Will the cities of tomorrow regulate traffic between individual buildings?

Make no mistake, bikes are much, much, better for urban centers than cars. But the overall problem isn't cars, it's individual traffic in densely populated areas.

Certain policy here in Europe simply assumes that people stay in their surroundings ("15 minute city") and rarely, if ever, visit parts that are farther away individually.

Public transportation, however, is naturally biased. It can be much quicker to get 10km north-south than 5km east-west, or the other way around, depending on the city. And, of course, public transportation is often lacking quality compared to individual traffic. (Taking a bike across a bicycle road vs. getting into a crammed subway train in July, for instance.)

Bike lane congestion is wholly unlike car congestion. It happens, and for busy routes, you do need affordances in the bike lanes and infrastructure.

That said, I've never missed a light cycle while bicycling, even in rush hour in major metro areas with many people on bikes. The busiest bike intersection in the world, in Copenhagen, devotes less space to bikes than most US streets devote to one lane of vehicle traffic.

Every single US city I have seen has ample space for bicycle infrastructure. Many have lots of roads with sufficient space for dedicated transit lanes, and bicycle lanes, and widened sidewalks while still maintaining space for personal vehicles.

Even if we take your hypothetical to the extreme, with only bicycles and no personal vehicles, every US city has enough space for the bike traffic on its existing streets.