I am currently in a similar situation as the OP. Our backend at work is Google; we use it for email, hangouts, and calendars, mostly. The first thing I did after installing Office 2010 is install the Google Apps Sync program for mail, calendar and contacts. We use this on all of our workstations, and it worked just fine on the new SP3. That being said, it's a desktop solution, but isn't that the beauty of this device: being able to run both?
I understand your point though. I too wish there were native apps for the most common Google features. What I like about my phone is that I have dedicated Gmail, Calendar, and Hangouts apps, and the notifications come through seamlessly. That being said, I already have my phone to do this, and I am more likely to have my phone with me than my SP3. This thing immediately replaced my (old) laptop and (old) tablet, but it will never replace my phone.
Now if I can only find the best solution for marking up PDFs...another thread, perhaps.