The club where I worked had carts, walking, and caddies. I never thought of pull carts; they weren't popular at that time. I assumed they were allowed, but maybe not. There was no time restriction on use, but a weekday caddie would need to be arranged ahead of time (safer to do it on a weekend as well). I wouldn't say anything was "promoted" although the pro shop clearly wanted everyone in carts, but that was a money thing for them. Members were going to do what they wanted, whatever the pro shop wanted. Carts were predominant, probably 90% or more. I don't know about the pro shop, but in the bag room we loaded everyone onto a cart, unless instructed otherwise. Of course, we recognized the very few adamant walkers and didn't load them. Sorry, Melvyn I'd rather not give the name but it was a good, old, prestigious club in the area, although not nationally or internationally renowned. Hope that helps!
And for the record, I'd like to know the rules for walkers as I am curious and learning about the subject (I assumed usually it was like this club, obviously not), and it wouldn't greatly influence any decision to play at a club or join.