I have got the reason of it.
As you said, you are using the some script from Google. However, I am in the most fantastic country, china, where the IP of Google is unreachable.
After use a VPN to make IP of Google be reachable, I can see the Search Box in all browsers in my computer.
You can check on this with FermiTeck.
However, if we can still make the javascript from Google available without connecting to the IP of Google. That may be wonderful. Just a wish.