As many have indicated, the only real problem with push carts is how they concentrate traffic, sometimes with paths of short grass are cut in the rough, sometimes around a bunker or between a green and a bunker. Sometimes those well treaded paths are actually a touch lower than the surrounding terrain and do get more occasional water. Concentrating all the traffic on a wet part of the course might cause mud, compaction, and thinning of the grass. I would never implement a 90 degree rule with push carts, but it might have to do with this. I would rather rope off certain areas and have people go around it, change traffic patterns. At our club we do require carts leaving fairways and into cart paths or rough areas some 20 yards prior to the greens. Nothing worse than having no grass and compacted dirt on fringes.
As an aside, with very wet fairways, damage can result from the wheels, but on normal days, the concentrated walking, carrying or pushing is far worse, in my experience.