[entire-name]

I think OP means they don't mind sharing their information with the search engine (be it Google, another engine that provides better results, or even a better Google in terms of results), _as long as_ OP has control over exactly what is being shared.

As an aside, I do see the trend for some companies to provide this control nowadays. Even Google is doing it (e.g. you can auto delete your information, or turn them off completely): https://myaccount.google.com/data-and-privacy

Of course, whether or not you believe Google is doing what you have configured in the backend is another question... and there is nothing anyone can do to actually make you believe it short of giving you complete access to the entire Google backend. Or is there a way to verify without exposing? Maybe an interesting research topic...