Is cycle sport encouraging uptake of cycling?

This almost completely stems back to the oil shortages of the seventies. In those times those European governments sought to alleviate the problems with a commitment to creating and promoting a culture of cycling and sustainable transport as a credible alternative to driving. It also didn't happen overnight and took many years to get right. England and the USA took the view that driving had to be supported as an absolute priority which led to a culture where driving is the norm and cycling completely unplanned for because it's seen as completely undesirable.

Also note that those more progressive cycling countries are devoid of any history of auto manufacturing.

In the US, the post-war highway system construction by Eisenhower was largely heavily driven/encouraged by the automotive sector. They already had the government contacts after their heavy involvement in manufacturing during WWII.