Brandon Dunes only allow the use of their push carts which are wider and have very large tires that spread the load in a positive down force (yeah, I called them). I haven't checked the Australian course but assume the same. The Villages have handicap golf carts that are allowed on the greens because their design also spreads the load.
I also prefer the judgement of superintendents at US courses, most trained at schools that teach Golf Course Management (Penn State, University of Colorado, University of Maryland for a few) then any posts here that search our minority examples to condone bad behavior.