We got some push carts at our club last season and I took one for a spin one day. I found that it is far more difficult to use than just carrying the bag. You have to traverse around the greens and greenside bunker complexes to get to the other side, as opposed to walking straight across the green when carrying. And it was real work going up hills. I never did it again. Using a good carry bag is way easier, IMO.
I agree with you regarding having to walk a lot more distance when pushing a cart. I usually push when there is a risk of rain because 1) my bag gets heavy carrying my extra rain gear and 2) I like having the umbrella attached to the cart. My course is quite hilly also and I still can't decide if it is easier pushing or carrying.
In the summer I definitely carry my bag.
My answer would be to use common sense. There are areas on the course where I work that even a push cart makes ruts if it's really wet. Another thing I see fairly often are visible push cart tire impressions on the fringes of some of the greens when it's really wet. Those people should be more aware of what they are doing and keep the push carts away from the greens.
At our course (and I think at most courses) we are encouraged to never push a cart between the bunkers and the green. Sometimes that makes for a long walk around the bunkers.