Bicycle infrastructure is primarily about towns and cities. So if you want this big push, you have to do it at the city level, but not just a bike path here or there, but redesign the entire city for bicycles. Have bike thoroughfares, make sure bikes are the best way to get around the center, and for any kind of traffic that stays within a town or borough. Cars are only for getting out of the city.
Many Dutch towns are increasingly redesigned with that idea in mind, most thoroughly perhaps the town of Houten[0], where the only way to get anywhere by car is to take the ring road, whereas bikes get quick shortcuts through the center. That makes bicycles attractive and keeps the pollution from cars out of the center. Travel within the town will favour the bike because it's easier and quicker than a car, which reduces the amount of car traffic, and reduces congestion and pollution.
That's the level at which you need to address this. Well, that and revoking any national helmet laws for bikes, because that's been shown to keep people in cars.
IMHO, this is 120% just a chicken and egg problem. Sure, nobody bikes that far to work, but if you build it people can and will travel large distances to visit just to go on bike tours. In my state, we have an annual event which attracts thousands in which people bike hundreds of miles from one side to the other. Rural highways are closed down/protected for bikes, but just for that one week.
In Europe bicycle vacations are pretty common, but it's very different from a regular commute. I did have one coworker once who biked about 50 km to work, but he was very much an exception. For most people, 10 km is a reasonable maximum.
That's the case for a lot of the trails in my state - there are big, rural, rail trail loops, but the suburban/urban connections to the routes are excellent commuting corridors.
Yes, they do. There is a whole series of trans-Europe routes, the Eurovelo network, and some of them are downright crowded with whole families in the summer.
I do some cycling every summer in Serbia and Bulgaria and my route sometimes intersects the Eurovelo 6. It is often a crap shoot if I am going to find accommodation in the smallest towns, because the rooms might all be booked by French and Germans on a cycling holiday along the length of the Danube.
One driver of that is rail trails.
In the densely populated Netherlands we are facing very serious gridlock problems now with cars clogging the highways and railways at full capacity.
We can't keep building new infrastructure (highways) at the pace we did before, because - besides being very expensive - we lack the space for it. New solutions may lie in promoting remote work, carpooling and shortening home/work commute distances.
I also suspect that bicycle infra will be developed even further (e.g. with bike highways) as they take less space and are less disruptive to the landscape.
You can build highways and rails in 3 dimensions. The bicycle simply cannot replace cars and trains in any meaningful way, despite the rise in numbers of single, childless urbanites.
But it kind of bothers me that Danes and Dutchmen seem to be rather disinterested in one another's achievements on this front. There is a lot of lessons from both sides, that never seems to be discussed in one another's media, to the point (anecdote alert!) where I've seen people in either country be surprised that there is a similar biking culture in the other country.
And people still love their cars in either country, even though I believe it's even more expensive to own a car in the Netherlands (and I thought Denmark was pricey!).