Yup, as per my note in another thread this does not work and it's pretty dumb to think it will work. Privilege covers you providing facts so they can give advice and them giving advice. It doesn't cover anything else and adding a lawyer to a thread like that is just sketchy and looks evasive.
“If simply having a lawyer present made a conversation privileged, then every mobster would pay one to follow them everywhere and sit in the corner of the room.” ~ Opening Arguments
I get the idea that if Google was delivering drugs to someone they would start off the conversation: "Are you a cop? Cause you have to tell me if you are!"
https://www.stroock.com/news-and-insights/u-s-supreme-court-... provides some more context - and speculates as to why the court declined to take the opportunity to definitively establish a standard here. It seems to still be an open question what the standard is!
You should be a bit wary of reading oral arguments as definitive, because the judges have a history of applying the more aggressive questioning to the side they ultimately agree with, just to make sure it's solid.
https://www.oyez.org/cases/2022/21-1397
From listening to the oral arguments, the court didn't seem to think that just CC'ing a lawyer automatically made the full communication privileged.