I'll take that bet too. Not a chance in the world a guy walking is going to play faster than me in a cart (cart path only or not). The only chance a walker has to beat me for speed is if he's carrying one club and named Usain Bolt.
Most of the time cart path only doesn't slow me down at all. I just put more effort into laying up close to the cart path. Every course I've ever played has at least 2 or 3 holes where it takes a walker longer to go from tee to green than it would take me to go from tee to green and complete the next hole.
When I first joined a golf club I didn't have a cart but it became very obvious that wasn't going to work if I wanted to play in the 12:30 money game. Some of the players had to leave at 2:30 to pick their kids up from school and a walker in a foursome of three people in their own carts was an anchor. To play in that game you not only needed a cart but also a fast cart.
Sure, I can walk that course in 2 hours by myself but not playing golf with a foursome. Even playing ready golf (which we all certainly did, sometimes to an extreme) there are still times where we have to wait, and can't just drive (or walk) up to a ball in the middle of the fairway to play our shot while somebody else is farther back putting us right in their line of fire.
Drove me crazy when I was the guy wanting to hit my second shot after hooking a tee shot, but couldn't because an overzealous "ready golfer" is standing right in the line of fire for my next shot, and in their own little world.