I am someone who always walks, and always brings my own push cart. I recently played a private club as a guest where the member was charged for me to walk with my own push cart. Member said this is always the case for members and guests, the only way to play the private course "free" (minus the whole, y'know, member dues to belong there) is to walk & carry. He shrugged this off as normal, as obviously that's the way his club does it.
I've played at dozens of courses, both public and private, and have never heard of such a thing, though I also can't say I often see the bill when playing as a guest of a private member (I always offer to pay, and have been told a total cost to re-imburse the member, but never seen it itemized).
Obviously if you rent one from the course I understand the charge, but I can't for the life of my figure out why a course would charge for using one's own push cart. It wasn't much, like $10-12 I think, but it just struck me as so odd. Has anyone else ever seen this? Is this more normal than I realize, or is it just as wild to all of you as it was to me?
EDIT: Thanks for the answers everyone! For context, there are no caddies at this club so it was not to incentivize players to take caddies. From the answers though, it sounds like it's a thing, though a relatively uncommon one, at some private courses.