A deeper tread depth will provide more traction for any tire and also do more damage to the surface it is rolling over. For instance, a golf cart or mower tire with a deep tread will leave marks on the grass. Many up-scale golf courses do not want such marks left of their grass and may not tolerate a tread depth of my than 4/32nds. The mowers used to cut greens use smooth tires. If you are stupid enough to drive on the green with your golf cart, the course manager will toss you out and send you a bill for repairs. Link to example products.
The tire tread used for greens and fairways is either a smooth, or a rib. These tires do not leave marks on the grass. What’s more, you want to look for R/S tires, or those with “round shoulders.” Round shoulder and shallow turf tires are for specialty applications—primarily for soft turf/golf course/sports field applications. Flat shoulders are for the tough stuff, not the greens.