There's one problem with a good tool kit: Theft.
If you use your moped primarily for joyriding, or commute back and forth to someplace where you can drag your kit in with you, that's fine. For those of us who use our mopeds to run numerous errands and attend various events, it's not always practical to bring a tool kit, or worse yet, an extra gallon of gasoline, inside.
The trick is to find a balance. Instead of an extra gallon of gas, I now carry a two-ounce bottle of oil (my fuel tank is smaller than a gallon) under my milk crate. It's a pain to get to without removing the crate and its bungy cords, so I'm relatively confident no one will mess with it. I limit my tools to whatever I can fit into my underseat tool kit, which is pretty small, because the average Joe is never going to think to look under there for valuables.
The cell phone, on the other hand, always stays on my belt.