Please don't characterize this as a political voices thing. Of course it's _related_, but this instance is about misclassification of private data, not silencing voices online.
They are a private company in the sense that they are not a government entity, and thus the first amendment does not apply to them. This is a different sense of the word "private" than whether their stock is or is not publicly traded on an exchange.
The nature of the problem (a big monopolist behemoth of a corporation is too big and powerful and lacks incentives to change) and the solution (only state action, regulation, mandates, i.e. something more powerful than Google must enforce Google to act) is by definition a political problem. This is like the epitome of what politics is for.
Also, Google is not a private company.