Lawns provide oxygen, filter rainwater, prevent soil erosion, and are better at absorbing / releasing heat energy than hard surfaces. But they take a lot of water to maintain -- in water-restricted areas, you're probably better off with more drought-tolerant vegetation that will have the same benefits.
The gas-powered lawn mower will always be bad for the environment.
What lawns provide over those are mostly aesthetic and functional, not ecological.
Getting rid of the lawn, and with it the lawnmower, is the clear ecological choice, even if not always the right personal choice. No koan or paradox involved.
Here in the Pacific Northwest many of us never water our lawns. They get plenty of water from rain except in the summer. In the summer they go dormant. That turns them brown, but doesn't harm them, and when it gets rainy again in the fall they turn back to green.