Improve search experience

From @katre in https://github.com/bazelbuild/bazel/issues/30598

> I had a chrome "Site Search" set up for the Bazel docs, using the URL "https://bazel.build/s/results?q=%s". It doesn't look like there's an actual search endpoint anymore: when I go to bazel.build and search it shows the results inline in the pane.
>
> Is there a way to bring back site search like I used to use?

The new search feels a little constrained compared to the previous solution. This issue tracks improving the search experience to restore previous search functionality.