xYz You're entirely misunderstanding what's happening and are drawing highly inaccurate conclusions about it. Your claims about this are unsubstantiated and untrue.
After I signed into my Google account, Google began auto-downloading without my permission all kinds of Google apps like Wallet, Gemini, Home, System Intelligence, etc., and asking me for install confirmation.
You chose to trigger this and it didn't do anything without your permission.
Google was able to track me across the web in my default Vanadium browser. I was getting popups at the bottom of Vanadium saying "Sign in to cnn.com [or whatever website] with google.com" with a button to select "Continue as [my google account name]."
You chose to use this and it didn't do anything without your permission.
After doing some troubleshooting, it appears Google installed cookies/trackers into Vanadium without my knowledge.
No.
When I attempted to login to my Google Play account on the 9a, Google sent a confirmation number from my old P7 to my P9a via Vanadium to confirm I was the legitimate account holder of the Google account, and this is apparently how Google was able to track me in Vanadium. I have since deleted all my cookies in Vanadium and haven't again experienced this issue.
No, none of that happened.
Once, when switching into airplane mode and turning WiFi back on, I received a notification that directed me to a Google “Pixel Phone Help” page with an article titled, “Keep your Android’s wireless connections on in Airplane mode.”
This has nothing to do with Google Play. GrapheneOS doesn't publish all of the user-facing Android help information on our site and change all the links to it. You chose to read the help info.
One additional thing I have not personally experienced, but came across on this forum is someone claiming the Google Gemini app is able to access the Internet even though network permissions have been denied.
That's not true and it's unclear why you're bringing up another inaccurate claim.
All this is very troubling as it seems Google was/is embedded into my phone in ways I never intended or expected and it makes me very reluctant to continue using sandboxed Play Services and just go back to using Aurora.
None of what you've posted is troubling or unexpected. You're making highly inaccurate assumptions about what's going on based on confirmation bias where you see a bunch of unrelated things as indicating what you think is happening when it isn't happening and does not indicate that.