It's not like getting rid of cars will make commute faster.
My friend (working in biology research) lived in England for a few years, and had 2+ hour commute by public transport. He was married, so both he and his wife had to find the jobs, and one of them had to end up with the large commute...
Since there is more space, you'd expect that the market would allocate more of it for parking where needs be and that you don't need the government dictating parking minimums.
Also, not all space is equal: even in the US, well-connected space with lots of amenities like the downtowns of cities is not infinite.
The real issue is people not knowing where to move. They go in google and search for top towns in the united states, and a bunch of city-data articles come up and next thing you know those drivable/parkable towns like Bozeman, Flagstaff, Bend, Grand Junction, Asheville (or Ashland for that matter) all become bogged down by massive population influxes. I'm talking like 2x 3x population boom in the last 5 to 10 years. There's no way these towns can't go from "Strongtown" status to homeless everywhere and clogged up roads if people can't get off google and find something new without google telling them something is hip.
The market allocates only as much as users can bear, which is about epsilon above the line of "for average user, the parking situation is so bad and unbearable that they chose to go without the service". Anything above it is money left on the table.
That's assuming there is a monopoly on providing parking spaces. Zoning notwithstanding, anyone can build a parking structure and rent or sell parking spaces. Tenants can choose whether to pay more for an apartment that includes parking or less for one that doesn't.
The total cost of a housing unit with parking wouldn't be be more than it is now because the cost of constructing a housing unit with parking wouldn't be more than it is now. It would just cause units without parking to become available.
Which would tend to increase housing availability, which would tend to lower prices including for units with parking. Because the people who want the parking spaces wouldn't have to bid against people who just want housing and don't need parking.
Fun fact - the city near me has a new zoning rule that limits the total number of parking spots in the city to something close to the current number of parking spots. If you want to build an apartment with parking, type find to need to tear down something with parking first.
So they're tearing down shopping centers, movie theaters, and grocery stores just so they can build apartments with parking. I have no idea where the people living nearby that don't have cars will buy food, but at least the new luxury apartments will have parking.
Japan has no street parking (and the occasional public lot run by the city) and I never felt this to be the case. You're right, there's no expectation of driving into a massive parking lot everywhere I go, but instead there are convenient paid lots at pretty much any trip destination. The only times these things get booked out is during big events when transit fills in the gaps anyway.
There IS more space in the united states, believe it or not.