Jim,
Early cart use did little damage because carts were either restricted to golfers with a medical excuse, or, the club could only afford a limited number of them.
As their number increased, something became obvious.
That the shortest distance between point A and point B became a dirt path from cart usage.
Hence, cart paths were initially used to divert traffic away from the heaviest areas of play, usually around the greens and tees and certain architectural features.
As caddies became extinct, and carts flourished, traffic and the problems caused by traffic increased, as did the impact to the golf course.
Enter rain and wet conditions.
When carts were prohibited from going on the golf course due to poor conditions, golfers complained that they paid their initiation and dues to play golf, and since caddies weren't available, and since they weren't going to carry their own bag, pressure began to build with respect to accomodating carts. In addition, as more and more clubs relied on outside outings to reduce financial pressures, the need for cart paths increased.
Hence, some clubs installed them to allow for play in inclement weather.
Clubs with robust caddy programs probably suffer far less in the way of damage to the golf course.
But, golfers today, don't have the same respect for the golf course, hence you see curbs, railroad ties, fences and ropes used to direct and control traffic.
I've seen golfers drive carts up to and on greens.
Golf carts, like water, are erosive if allowed to run on the same area, over and over again, day after day. And, the areas are almost always at the green, tee and near unique architectural features in play.
So, the long answer is YES, carts, but no cart paths, harm/ruin golf courses.