>If my friend on the other side of the country says "I woke up at 9 this morning", I have a pretty good idea of what that means. If we used this new system, i'd have to mentally translate.

Mentally translate to what? If your friend tells you they woke up at 9 this morning in a world with one timezone, you know exactly when they woke up. 9.

Currently you have to know where they are currently located, and you are currently located, to figure out the time they actually woke up.