It isn't the world we live in. I have no problems wearing my most expensive watches in public in Oklahoma City or Tulsa. No one in either city has ever tried to rob me. Even in cities like Fort Worth, San Antonio, Colorado Spring, Santa Fe, etc. I have never had a problem.
I know that we have bad folks living everywhere in the world, but in most corners your odds of encountering one is slim. It is only when you travel in extremely touristy places that your odds of being pick pocketed or robbed go up. And it isn't because London is inherently awful, it is because a place with that level of tourism attracts thieves. Thieves who know that there is a large number of tourists who will only be in the city for a day or two who make for easy marks.