Safari isn't available as a web browser

I just realized you mentioned your phone. This article discusses searching for websites on iPhone.

Search for websites in Safari on iPhone - Apple Support

iPhone/iPad

Safari  icon missing?

Safari is maintained as a part of iOS. There is no self standing version of Safari.

Go step by step and test.

1. Use the App Library to find your apps

     From your Home Screen, swipe left until you see the App Library. 

     Your apps are automatically sorted into categories.

    Organize the Home Screen and App Library on your iPhone - Apple Support

2. Make sure that Safari is not restricted.

    Settings > Screen Time > Content and Privacy  Restrictions

    Enter Passcode if you have set it up

    Allowed apps > Safari

    Turn it on.

    Use Screen Time on your iPhone, iPad, or iPod touch - Apple Support

3. Reset Home Screen

    Settings > General > Reset > Reset Home Screen Layout

    Return iPhone settings to their defaults - Apple Support

    Return iPad settings to their defaults - Apple Support

macOS

Click Launchpad in the Dock.

Click Safari

When it appears in the Dock, right click it.

Choose options and then "Keep in Dock".

Hello,

I think you may be actually talking about the default search engine to use when searching in Safari. Safari has never been a search engine but has been and still is a web browser. You can enable Safari Suggestions which include suggestions from your default search engine and more which may be what you are looking for.

Change Search preferences in Safari on Mac - Apple Support

"Safari Suggestions

Safari Suggestions appear as you type in the Smart Search field. You might see suggestions from the internet, weather conditions, information about flight status, recommendations from Apple Music and the App Store, movie showtimes, and more."

I hope this helps.