More hidden maybe, but definitely more logical.

More logical to use a textbox typically reserved for navigating away from the current page, to instead search that page?

Thanks, was going to answer the same: nor the "address" text field nor the "share" button indicate actions to be performed inside the page; the logical idea is that first one is to go to another site, the second one is to transfer the current page to another app/person

I have to admit I don't see where to put the "search inside the page" feature, definitely some ux trick to be found