Grass when not done correctly, can definitely be worse than just going with turf.
I remember when Cincinnati's Paul Brown Stadium was built, they had years of issues with the grass. It tore up easily and resulted in a bad playing surface.
The Tennessee Titans are experiencing the same thing at Nissan Stadium in Nashville. Tennessee State and the Titans use the stadium and the field is often a mess. They've not held the high school state championships there because of the wear and tear it would do to the field. The Titans have proposed a new turf stadium, and there is already talk of moving the high school state championship there.
Going to grass isn't a simple solution on its own.