Trust and user experience are two different things.

I would trust them not to spy on me and not to gather data for behavioural advertising whether they had no banner, a banner, or an autoplaying video every time you start Docs. I just wouldn't be equally happy with the product in each of those cases.

What bothers me is not the banner itself but their reaction to it.

Someone who handled this better recently: Mozilla got user feedback on their AI integration, and added an off switch.