It's not unrealistic to not carry a phone everywhere. Do you really need to be reachable while you're at the grocery store? Unless you're on call or a firefighter, there are very few cases where you must be reachable 24/7. Try it sometime, you'd be surprised.

Carrying a phone at all times is more a matter of social pressure than anything else. If you get into a real emergency situation, you can just ask someone if you can make a call from their phone -- after all, the world is full of people walking around with phones in their pocket.

Leaving your phone at home most of the time is a little inconvenient, but it's certainly not unrealistic. It's just that most of people are happy to trade privacy for convenience.