The search bar on the home screen referenced by OP is a Pixel launcher exclusive, which is the default installed launcher on Pixel devices. You can't remove it. You can install an third-party launcher however. You can also use a widget provided by a browser app to put a search bar on the homescreen (all major browsers have such an option).
For the most part, my webviews are in FF (selected as my default browser), but it depends on what app launches the webview. Sometimes it defaults to Chrome.
I was wrong about the web views, I didn’t realize they updated because it seems like app developers can choose to use specific web views instead of the system one.
Also wrong about the launcher, I’m relatively new to Android and hadn’t seen any settings for it.
For the most part, my webviews are in FF (selected as my default browser), but it depends on what app launches the webview. Sometimes it defaults to Chrome.