Option to set default browser does not appear

On your iPhone, do the following:

1) Go to your Gmail app

2) Click 3 horizontal lines in top left corner

3) Scroll down and click "Settings"

4) Click "Default apps" at the bottom

5) Deselect "Ask me which app to use every time"

6) Select your preferred browser

7) Repeat #5 and #6 for your map, navigation, and calendar selections as needed

THEN, on your iPhone, do the following:

1) Go to your YouTube app.

2) Click your profile photo in the top right corner.

3) Click "Settings".

4) Click "Default apps" in the top section.

5) Verify that your preferred selections have been retained.

Have a great day.

Safari is a Web Browser, not a Search Engine. You will still use Safari on your iPhone if you choose a search engine. What Apple did with the new release this years is let you choose which Search Engine Safari engages when you do a search and the options are:

Google

Yahoo

Bing

DuckDuckGo

To choose which one you want, go to Settings > Safari > Search Engine

"YouTube and Gmail constantly asks me to set a default browser. I would like to keep using Safari.

You kind of did.

Again, Safari is not a Search Engine, but I don't know why you're getting these requests to set a web browser. Your original post says you want to use Safari as your Search Engine. Since Safari isn't a search engine, instead of "do not ask," perhaps turn that off and select Google since you are using Google Apps when you are being asked.