On the other hand, the reduction in cars due to people switching to cycling makes the infrastructure incredibly cheap.

Look at the video in [0]: how much space would you need if every single cyclist was driving a large SUV? Look how smooth the traffic flows through the intersection, how many flyovers would you need to achieve this with cars?

Yes, cycle infrastructure does indeed take up a nonzero amount of space. But it easily pays for itself by reducing the need for far more space-consuming car infrastructure.

[0]: https://www.youtube.com/watch?v=2RQrKP9a0XE

> the reduction in cars due to people switching to cycling makes the infrastructure incredibly cheap

This is a logical leap of enormous magnitude.