Unfortunately not. HE has some performance issues (that might be solveable with time) but the primary issue is that Google do actually want to see exactly what you're doing. It's important for basically everything Google does that makes it money.
Furthermore, the privacy concerns are nowhere near great enough. All of the mainstream privacy concerns have come from Google releasing your information to others (Google Buzz, that recent ECHR case etc) and not Google keeping information themselves. Furthermore, since I assume that what you're really talking about here is emails, it would be impossible to practically implement a HE system for emails that would allow for searching (because one could derive meaning from emails by sending them the right queries). It'd be no better than client-side encryption, and no one really wants that.
No, the people for which this is really useful is large non-IT companies like Boeing or healthcare companies etc. They have huge data processing or storage requirements, and frequently have to run around hoops in order to ensure data is not transmitted in a way that would breach commercial agreements or confidentiality requirements.
The problem is that frequently these workloads (especially for companies like Boeing) will involve much data, and HE tends to come with a large size increase and performance drop - so it's often too costly, which is a shame.