The average backyard size in the US is 2,164 m2, which would support a family of 4 in your scenario but obviously suburbs full of backyards don’t actually feel very nice and green. Central Park is so amazing because it is adjacent to high density NYC. You have the amenities of a city and a large greenspace. NYC is hyper dense, but that is closer to optimal than rural Georgia.
Your Central Park example also points out another issue, which is money to maintain these greenspaces. At a fundamental level, it is a case of not having enough people to pay taxes (or whatever) for something even 10% as nice as Central Park. So you are back to what PartyOperator said, either bare grass, or wild forest. Another aspect is that people move to cities for a reason. Living in higher density allows for much greater and easier economic activity. Work From Home is nice for the few types of jobs that can do them, but most people need to go to work and doing that is only feasible with density.
Looking at your Swedish community example, that is only possible with density. You can even see from the overhead map that it is mostly smaller, hyper local parks and other mixed use areas, with a larger more unmanaged greenspace around the outside.
You can’t have your cake and eat it too. You can’t have the density of a deeply rural community, and also have huge, managed greenspaces and local shops. You should look into Strong Towns [1], that has a lot of good actual city planning information and what makes a happy, vibrant community.