I disagree with your "probably shouldn't be wearing such clothes outside anyways" thing - I know I'd prefer a bike I can just hop on in whatever I'm wearing and simply not even have to think about it. In fact, I just bought a bike and went waaaaaaaaay out of my way to get a chainring guard as it didn't come with one (let me tell you - it's a PITA to find a chainring guard for a 50 tooth chain).
In regards to "cleaner than a chain", while it's a little bit marketing hyperbole I think it's still true -
1. If I put my bike in my car, I don't get chain grease on stuff (there's chain grease on the backs of my seats for example as I have to wedge my bike in between the front seats and the back seats).
2. I don't get anything oily when I oil the chain - don't have to take it outside or to the garage. (Unrelatedly you don't have to oil a belt at all)
3. No matter where my bike ends up, I just don't have to worry about getting chain oil on anything that might fall against it, press against it, etc.
Now my personal opinion is that I'm going to take a "wait and see" attitude about whether using belts on a bike rather than a chain works out. It's new, and I'd rather not be the first in the pool, so to speak (tried it once with a bike already, could have worked better). But I do think it's an intriguing idea - in theory far better than a shaft drive (no efficiency losses (well we'll see), no grease to add, no (or negligible) extra weight, and a simple enough change that your local bike shop mechanic isn't going to look at it and say "we don't even know how to work on these").