Your cycling fallacy is…
“Nobody would choose to cycle if they could drive instead”
In places designed primarily for car use, it is no surprise that people choose to travel by car. And for longer journeys many people will choose to go by car, or by bus, train or air.
However, one big problem with the car as a mode of transport is that too many people using them for short journeys leads to traffic jams and poor air quality. Nobody benefits from this (with the exception of the oil companies who profit from people going nowhere fast).
But when a town or city is designed to enable cycling, then it becomes the obvious choice – easy, convenient transport for local journeys, such as to schools, shops, etc.
In countries such as the Netherlands and Denmark, where cycling is catered for as a serious mode of transport, people choose to cycle because it's safe, convenient and pleasant – and journey times are reliable, as they won't get stuck in congestion. People in those countries can drive if and when they want to, but because driving isn't the only choice for travel, as it is in many other countries, they choose to do so less often.
Everybody benefits from this increased level of cycling – even those who do need to drive, as it means that they're not sat waiting in traffic which is made up of people using a car for short local journeys.