On the other hand, going to another company may or may not help that much either, and for two reasons. First, depending on where you are inside Google, the technologies which you have picked up may not be all that useful outside of Google. More importantly, and you've pointed that out for yourself, if you don'y have habit --- and the curiosity --- to deeply learn new the technologies you are working with, you're probably going to struggle wherever you are.
Spending a month for each new technology that you think you need to learn is not going to be enough to deeply become an expert in any of them. Also, it sounds like you have some not-so-great work habits, such as not doing the best possible job you can with any assignment you have been given. Shaking those is also going to be helpful for you, no matter where you are.
So... here's what I would suggest for you. First, spend as much time working on yourself as you so working on "new technologies". Try reading books such as Steven Covey's The Seven Habits of Highly Effective people. I found listening to books such as Brian Tracey's Success Audio Tapes to be helpful in my early career. If you are more spiritually minded, some of Og Mandino's books are old classics.
Secondly, try to bump your performance review ranking at least one level each cycle, until you are getting strongly exceeds expectations. Not because you necessarily want to stay at Google, but because of the self-confidence that this will hopefully help you gain. And tell your manager that (a) you feel that you've been slacking, and (b) that you want to do better. If your manager is any good, they will want to work with you. Try to get promoted at least once more at Google. Why? Because if you are going to try to strike out at some other company, people who understood Google's performance levels will not necessarily be impressed if you have been at Google for six years, and are still a SWE III.
Finally, once you have a string of good ratings so it will be easier for you to try moving to another teams, you might want to consider working at some team which has contact with outside customers, especially in the Cloud PA. This will give you a lot of contact with external technologies, since customers will use a variety of different software components.
Please do keep in mind, first and foremost, that it's all about how you can add the most business value, no matter what company you happen to be working at, and no matter which customers you are trying to help. It's that work attitude which is going to be the most important, which is why I started this by suggesting that you work on your soft skills as much as your technology skills. In addition to listening to various success tapes while I was commuting to work, I also made sure I knew how to read a balance sheet and monthly income/expense reports. I also took supplementary classes in management (which my employer paid for) for subjects such as "Law for the I/T Manager" at the MIT Sloan School. This is all going to be super useful, especially if you think you want to leave Google; at large companies, you can get by just being a technology specialist, but if you are working for yourself, or at a small company, being a well-rounded employee who can understand various business and legal issues will stand you in good stead.
The bottom line is you need to wake your curiosity to learn as much as you can in a wide variety of subjects; not because you want to get a good/interesting job elsewhere, but for its own sake. And you need to develop good work habits and have the internal drive to do the best that you can no matter where you are. Jumping ship to some other company isn't going to change who you are; and you may find that it is much more about you than your environment.
If you want to discuss this more, look me at at Google and I'm happy to chat some more. My ldap is the obvious one at google.com.