On my Android phone, it is AFAICT impossible to remove the Google-branded search bar from the top of every page of the "desktop" (or whatever we call the basic app launcher screen on phones). On my Windows 10 computer, it is trivial to remove Cortana from my taskbar.
I've never heard of a device that wouldn't even let you install a different launcher.
You are aware that your "desktop" is a separate app right? And like your keyboard, you can install replacements. I use this [1] launcher and it lets me turn off that Google search bar.
I don't mean to be condescending, but either you have an incredibly locked down device, more so than I've ever heard of, or you're just understandably unaware of that aspect of Android.
Thanks for the link! I honestly had no idea that such a thing even existed, and I spent a long time trying to figure out a way to just drag the search bar off the screen like any other widget. Though still sort of a shame that I have to install and trust yet another third-party app just to do such a basic operation (I suppose it's some consolation that Android at least ships with a flashlight app these days).
If you bought a phone that's so locked down, you can't install a different launcher, I am sorry. It's actually really sad how locked down some android devices are, and part of why I do extra research when buying my phones...
Working backwards...
Nexus 6P
OnePlus One
Nexus 4
Some samsung POC
Original G1
(nokia pre-smart phones)
Even the samsung was able to install a 3rd party build of Cyanogenmod (checked before buying it)... every phone since the G1 has been android, and until the Nexus 4 I ran cyanogenmod... the N4, OPO and N6p are all running stock, since they include wifi hotspot in the box.