> whenever a user, including a user in private browsing mode, visits a website that is running Google Analytics or Google Ad Manager, "Google's software scripts on the website surreptitiously direct the user's browser to send a secret, separate message to Google's servers in California."

Which part of this indicates that the Chrome client contains code giving Google special privileges to track users in incognito mode that other websites don’t have access to?

>Google's software scripts on the website surreptitiously direct the user's browser

is referring to JavaScript. Any other analytics platform could (and does) do the exact same thing.

You claimed something was there, the burden is on you. The allegation is Google, through a combination of data collected from you Incognito browsing session and non-incognito data collected from other Google apps that happen to be running on the same device could be enough to deanonymize you. There is nothing at all alleging they use Chrome. Even the most egregious X-Client-Data header is called out as specifically as not sent in Incognito.