Technologically this is straightforward: it uses a proxy server that sits in between you and your actual mailserver.
I think the privacy concerns of having your mail (potentially) available over yet another server in exchange for modest convenience makes it unlikely that I would use this, but I'm sure many will find the trade-off acceptable and desirable.
You're doubling your surface for anything unethical, too. It not just the NSA, so it really makes no sense to take additional risk in this regards. You are adding an [or] operator, one that can lead to a complete failure mode. This is the opposite of risk diversification. Unless I'm missing something.
Yes, Google has your email, but not your credentials. A breach of your email exposes all the email you have now. Bad, yes. A breach of your credentials exposes all the email you get until you change them, and if you don't know to change them...
Before every email as yourself "if this email went public, would anyone care?" If I ever answer "yes", I don't send it. I either call or meet in person assuming we're not too remote.
Yes, I would never want them (or virtually any third party) to have the ability to access my inbox, much less stand in between my mail provider and myself.
I think the privacy concerns of having your mail (potentially) available over yet another server in exchange for modest convenience makes it unlikely that I would use this, but I'm sure many will find the trade-off acceptable and desirable.