> and trades open green space parks for shorter, 1-2 story buildings.
Misleading. American cities have lots of short buildings, but they also have more land to put stuff on, be it trees or buildings.
> The lower density means people are further away from the few parks that do exist too.
In some parts of the US, this is absolutely true. In others, it absolutely is not. Silicon Valley is easily the worst place I've ever lived in this regard. As a Midwesterner, I had never lived more than ~400m from a park, even in the suburbs. In Santa Clara, I'm more than 1.5km from the nearest public park.
Most Midwestern and Eastern cities do not match the "sprawl" archetype that most techies associate with America. Look up "Urbs in Horto."
> American zoning forbids taller ~6 story residential building
You're allowed to build up to 5 stories out of lumber. So a common archetype for American apartments is a first floor of retail space made of concrete, which serves as a base for 5 stick-built floors of apartments. That's where 6 floors comes from.
But the statement on its own is false. You're certainly allowed to build taller, permits permitting (heh), it's just that 5-over-1 is a local maxima for cost efficiency vs. likelihood of getting permits.
There are very few cities with laws on the books that prohibit building taller than 6 floors. The issue is that you have to get approval from the city to build things, and residents get angry when you try to build high-rises. So, permits get rejected.
This is an important distinction because the route to change is wildly different. We need community attitudes to change about dense housing developments, we don't need to change any law that's currently on the books.