Ok lets be honest here. The average user does not care or know what it actually means to share your data (even if not with an AI), they just want a service that magically works.
There'll also be a portion of people who are of the opinion they have your search history anyways and its already shared with the AI, just not specifically used for you so you might as well share to get the use out of it.
In the end the amount of people who care enough about their privacy to say no to this is infinitely smaller than the masses who do not care until its too late.
I say this as someone who just allowed any sharing and storing with google cause of convenience just a few months ago who has now taken a huge leap in transitioning to as privacy minded as I can (still working on switching from ecosia browser to floorp and windows to linux, but most of my services like email, maps, navigation etc are already more privacy focussed)