Culture might be the reason the Netherlands started down the path they're on today of building more bicycle infrastructure and we haven't, but...
The thing is, culture influences infrastructure, but the reverse is also true. Bikes in the Netherlands are a way of life partly because it's very safe to bike to your destination there! You don't need Dutch culture to get to that point, look at Vancouver: https://youtu.be/N9JHfFutvik
You're right that at the moment, bikes in the US are mostly considered to be recreational, and as a result they're not treated seriously as transportation. But I think people will get to where they're going however is most convenient, and as a result we can get more people cycling by making it more convenient.
(You can count me as one of those people; I have little interest in cycling for recreation, but I would absolutely bike around the city if it were safer. Given the limited reach of our public transit, it would honestly be the quickest way to get where I need to go most of the time.)