If you don't know how all this works, which is true for most human beings, their privacy policy might give you the wrong impression.
That's not what it says. It says the header won't contain PII, which is true. It can be linked to PII, but so can literally every bit of information you send to Google while logged into or otherwise using their services. A disclaimer to this effect would not have any purpose.
I don't know about your PII thing, but it's personal data under the GDPR.
That's their policy towards privacy, you don't have any. For some reason I can't fathom, you claim mentioning this in their privacy policy "would not have any purpose". Instead of honesty, their privacy policy is a wonder of public relations where it seems like they care deeply about protecting your privacy.
Under this formulation, Google's policy is (presumably, lacking any data to the contrary) honest with respect to this value.
This is key. If you subscribe to the "how it could be used" version, then even say possessing an android phone would be a violation of the privacy policy. Which is absurd.
Google's policy does not tell the user how her data will be used by Google's customers. The policy states Google will use the data to "provide better services". That is deliberately vague. That is the "purpose", but how exactly is the data used to achieve that purpose. There are no specifics with which a user could object.
Google does not only serve the search engine user, the email user, the YouTube user, etc. Its business is not free services. As such the policy is misleading as to what are the "Services" it may use the data to improve. Google's business is providing online ad services.
The truth is that Google collects data to provide better services to advertisers. The policy reads as if it only collects data to provide better services to users. The "free" services are just bait to draw users in. The data is collected to improve online ad services.
I understand that that is what you believe, but I do not think this is factually true about the data collected from this Chrome header. I believe that Chrome team collects it in order to understand the impact of Chrome experiments on performance.