Simpler and more reliable solution for bug 66506

This reverts change Id8b0d13 and also removes one line from
SearchOverlay.html (thereby avoiding the fulltext search when
Javascript is active). To enable the fulltext search for the
'search within pages' button, we have it re-add the input into
the form.

Bug: 66506
Change-Id: I4d65dad56e79cb7784dd9d64aa20bdac089dff4b
Merge "Simpler and more reliable solution for bug 66506"