It wasn't just showing off land. It was showing off you had land that was completely unproductive and could even afford staff to maintain this completely useless plot. It's why they became increasingly large and ornate gardens.
It then took this association with rich and better and was used in the US to denote affluent and therefore segregated housing developments.
You see random farm houses and country estates with acres of field that get nothing but mowed they're essentially preserving resale value.
Generations ago those field would have been allowed to grow over because they could be cleared again on a whim.
These days you need six figured of engineering and permitting to clear acreage and ain't nobody gonna do that without a commercial use that can justify it.
Grass doesn't prove it's fertile before modern fertilizers. Natural grasslands are often resource-poor places that most other plants can't tolerate. We sometimes replace one for the other because our main crops (wheat, maize, rice, sorghum) are all grasses, but e.g. Versailles probably would have had more forests, lakes, or a vineyard if it didn't have the gardens and lawns.
It then took this association with rich and better and was used in the US to denote affluent and therefore segregated housing developments.