I think the idea is great - if (when?) this were completed, many things would be simplified. I think many people would distrust the idea, though. Some people wouldn't understand the concept of telepresence, and regard the robots as dangerous and capable of "disobeying their masters." Those who understood might still be uncomfortable around a machine capable of engaging real humans in combat (even non-lethal combat). Machines can malfunction, even if they're under human control. Imagine accidentally firing the pepper spray fogger when you were attempting to shake someone's hand (or pick up garbage, or whatever this machine would be capable of). If perfected and given an image that was trustworthy, I think this could be a huge step.
The first of many obstacles, though is the question of "how." Do you have any ideas in the way of technology that would be capable of this? I assume that a video camera and microphone would be integrated, as well as a speaker for your voice to emit from. And a simple build could easily be modeled after a remote control car - although I don't know how you would handle the problem of range. To be in one city controlling your robot in a different city would be a very impressive feat. With wireless signals and such I'm sure the possibility exists (and probably has been acheived with different modern technology), but I wouldn't know how to go about it.