There's a lot to be said about this approach (and I had a good conversation about this with someone I know who works at Google and leans conservative / Republican but not alt-right, and is unhappy that internal social media is tied to business-related communication - blocking someone who posts a lot of partisan political stuff apparently means that you can't communicate with them in an emergency, if they maintain one of your dependencies, which seems like a pretty bad design).
But I think there are two reasons it's not a completely simple solution.
The first is that Google, as the article says, encourages you to "bring your whole self to work." Heck, Google even encourages you to bring your dog to work. Google encourages you to stay for lunch and probably dinner. Google encourages you to form bands with your coworkers, go on hiking trips with your coworkers, etc. There's a lot of deliberate attempts at promoting social cohesion beyond just normal traditional professional relationships, and by all accounts, that's a lot of why Google is as successful as it is. Requiring "Don't talk politics at the workplace" involves not talking politics over meals, on weekend team trips, and so forth, which is unlikely to be enforceable and also seems like it would hurt the genuine social bonds these things are trying to promote.
The second is that "politics" includes things like "Is it valid to be trans" or "Are certain races inferior to others." I'm not going to deny that this is "politics;" the generalized question of whether whole swaths of people are deserving of recognition under their claimed identity has been part of politics since basically the first empire captured its first nation. But it's unavoidable for a large company to pick an answer one way or another. A hypothetical Google running at the time of Cato the Elder, with Roman and Carthaginian employees, could not avoid the "political" question of whether Carthage should be destroyed and its residents sold into slavery.