2 Reasons I would never consider using this:
1. I don't want anyone to have any idea "how busy I am". Maybe I'm old fashioned, but the size of my inbox, like many other things, is no one's business but mine.
2. This effectively makes the sender the gatekeeper for deciding what's important. I want to be my own gatekeeper. I would much rather have everyone send everything when they would normally send it and let me decide when or whether to read it. I can't think of any good reason why losing control and losing data would be advantageous. I'd rather figure out ways to manage my time and my inbox than wonder what I'm not receiving because someone else made a decision "on my behalf".