IMHO, golf is a sport that involves the small amount of time spent on the course surveying the next shot, addressing the ball, and striking it. How you and your clubs arrive at each shot (via cart, legs, caddie, push cart, etc.), how you spend that time between shots (appreciating your surroundings, evaluating the design, taking photos, bs-ing with your friends, drinking, smoking, or whatever), and what you wear when you arrive for each shot (jeans, cargo shorts, slacks, golf shoes, sandals, etc.) seems more or less irrelevant to me. I prefer walking in very casual apparel and appreciating my surroundings between shots, but I play with others who prefer whizzing around in carts stocked with Bud Lights... We all love the game.
If you take away the carts, I think they'd keep playing. It's really the sport (hitting the ball and scoring) that attracts us most. Riding, walking, gambling, drinking and the rest are just icing on the cake... Pick your preference.
That said, calling golf and cart golf 2 different games seems ridiculous.