people commuting in is a large part of the work force in cities, tourism and generally shopping are important for their economies, also a lot of that parking space is needed by the residents because chicken/egg they all have cars too.

But why so much space... Why no high parking garages, or deep underground ones.

Cheaper, easier. When the land becomes more expensive, it would be built up with offices and the parking space either dissapears or goes underground or high up.

Why not? America is large, land is not really in short supply, except in dense cities.

Because taking up more land is cheaper than building up thanks in large part to the various rules and regulations of municipalities and states. And in many places a parking garage is valued and taxed as a structure whereas a parking lot isn't. Prefab concrete parking garages are a thing the same way that prefab steel warehouse buildings are. The structure isn't that expensive over the life of the structure. It's the latent cost of building and owning a structure that pushes the economics in favor of parking lots.

Ironically in a "you are the problem" sort of way, the demographics doing most of the complaining about parking lots and a general lack of density are the same demographics who are in favor of fine grained management and regulation of what people do on their land that causes this situation. You reap what you sow.

If you own a parking lot that is too small, you can go around booting or towing cars that "don't deserve" to take up your limited parking space. That can get you an awful lot of money, which can get you an awful lot of lobbying power.

Big structures are expensive. Who is going to fund them save for the people who are incentivized not to?

They do need the space for parking, because they have to drive everywhere, because everything is so far apart, because so much space was given to parking...

Everything is so far apart because America is very large, not because they spaced things out with parking lots. But it does sound funny to make that the reason.

"Yup, this big country was spread out by parking lots."

We’re not talking about why Seattle is far from Miami. American city sprawl is largely a recent phenomenon (post WWII) and was enabled by the rise of the personal automobile. Even the sprawl-iest of cities (Houston, TX) used to be respectably dense and walkable. The need for a car to achieve basic needs (groceries, school, work) is a phenomenon well less than 100 years old. America has this problem in part because it’s much younger than a lot of European cities, but it’s certainly not purely a function of topography.