Do they require all your golf bags and shoes are the same color too? I've never heard of a club requiring the rental of a push cart for conformity reasons. I'd ask if I could pay the $6 and use my own cart anyway as most rental push carts are broken or very cheaply made.
I love my Sun Mountain push cart and would use daily BUT my club doesn't allow personal carts so I end having to use one of theirs which isn't near as good. They say they want conformity at the club (translates to revenue $6). I walk 50/50 depending on whom I'm playing with. Much better exercise in steps. Avg walking is 12,000 steps as compared to 2,000 riding