I prefer walking but when required, a cart is fine. I walk about 70% of the time. Eschewing a cart saves a fair amount of money over a golf season, too!
Your experience of walking too fast was interesting. I have never noted my hearted pumping or that "exercise rush" you described, except after climbing a steep hill. My thought is you are correct about walking too fast. Don't stroll along but golf isn't a race, either. I find when I play alone my tendency is too move along too fast and not take a moment now & then to rest. My legs get a bit tired toward the end of the round and that typically results in poor swings down the stretch.
As to having too much time to think, that has not been my problem. I like walking as it gives me a break between shots. I can cool down from a poor hit as well as bask in the warm glow of a well struck shot for a while when walking. It also improves my game. First, it is easier to keep track of your ball. Rather than shoot off to another player's ball, you walk directly toward your own. There are times in a cart that I get disoriented from driving to other player's shots. When it is my turn who the heck remembers where I hit it? Second, while walking to my ball I feel the wind direction, note whether the next shot is up or down hill and generally think about the strategy for the next shot. Zooming up in a cart doesn't give me proper time to think. Just instinctively hitting a shot may help you but it doesn't work with me.