Should Google search index be forced to be public too?

Honestly, yes it should in some form. If their index contains the actual data from the sites, and they are making that information public in one way or another, then it should be available as a downloadable dataset.

How far can we take this?

Should Boeing airplane designs be public domain since the underlying math is public domain?

I don't think that slope is as slippery as you think it is.