I wonder how much of that association is precisely because of this pressure/requirement in a lot of communities to maintain grassy lawns, rather than anything innate about grassy landscapes themselves.
I think you are on to something. I grew up in west Texas and while there was no canopy there were/are vast untouched lands of native grasses. I still get the benefits described when I go out there.
If you follow with some of the "ancestral genetic memory" theories, this makes sense. Our species came about in a grassland punctuated by rocky outcroppings and trees. Trees and rocks often meant safety from the predators below. Open grass meant exposure on all sides. We don't really think of ourselves this way anymore, but not that long ago, humans sat more in the center of the food chain that we currently do.
Around 800 square meters of lawn. Every week, or every second week, depending on season and rainfall, I plug in a mower and mow it, a bit of exercise and fresh air thrown in for free. If for some reason I skip a turn, grass and weeds grow a little taller.
Where's the stress?
Not if you just mow it and let whatever grows grow.
I am a borderline lawn nut. After spending a lot of time getting the lawn well established, healthy, and vibrant, the on-going maintenance to keep it looking very nice is not particularly onerous. I doubt it's even 25 hours of total labor across the year, all told (for a 6500 ft^2 of lawn city lot).
I'm the opposite. I don't like lawns, not just for the looks but mainly because as far as biodiversity goes a trimmed lawn is almost dead. So I converted the lawns which came with the house into meadows. Nature thrives, and as far as maintainance goes it is also beneficial: depending on conditions I'm mowing twice or ideally just once a year. Even with a scythe it takes less time than standard lanw maintainance.