I moved away from FF to Chrome many years ago. I don't remember ever seeing that message. But that may be due to the way I manage my computers.
First, I never log out or reboot, except when I get an update. When I am done using my laptop, I simply close the lid. That puts it in hibernate. When I want to use it again, I just open the lid and press the start button. Everything starts back up the way it was, before I closed the lid. That includes Chrome with all my previous tabs and a Google login.
To satisfy Google, I have a couple of ".gmail" accounts set up. However, I never actually use them. I send them a message, now and then, just to make sure they are working. But that is the only activity they get. They don't even get spam. Once in a while they get a message from Google telling me about the terms of service. I usually just ignore it and throw it away. One of "gmail" account is logged in to Chrome and stays that way.
I also have a Windows 10 desktop that uses Chrome. That system I never shut down. If a update comes along, it will get installed and the computer rebooted. But when I log back in, Chrome comes up again, the same way it was before the update, and I just move on. No Google login message. I use the same Google account as my Windows 11 laptop.
And, in the end, if any web site requires me to do anything, like approve cookies, disable something, or log in, I delete them. They have nothing that I can't get from another web site.