Taller Grass Has More and Deeper Roots
Years ago I observed an experiment where lawn grass was grown in pots and cut at various lengths. After 6 months soil was washed from the roots and dry weight was measured. The grass cut at 2 inches had twice the weight of roots as the 1 inch cut.
Then a year or two later I observed another experiment where the grass was grown in soil with a glass wall so the roots could be observed in an underground cellar. You guessed it, the root system depth was in direct proportion to the mowing height. The two inch cut had twice the depth of roots as the one inch cut.
If all other maintenance practices are equal, the higher cut lawn will always look better than the one cut short. Higher cut is more important as weather gets hotter.
But, you say, golf course grass is cut short, and it looks great. And golf courses spend a lot of time and money on extra fertilization, pest control, aeration, top dressing, and over seeding to keep the short cut grass looking great.
I am currently cutting my lawn at 2½ inches (after mowing).