Hi roundabout,
welcome to the Apple (user-to-user) Community,
Re: "On my iPhone 13 Pro Max I'm missing the default search engine button in settings".
Wondering which iOS version is installed on your iPhone:
If you haven't already:
You might want to try updating to the newest iOS 16 update:
About iOS 16 Updates - Apple Support says iOS 16.4.1 is the latest
or
If you have already updated to the latest iOS:
Sometimes restarting your iPhone will restore order if something is not working as it should: How to: Restart your iPhone - Apple Support
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Apple support's article:
Customize your Safari settings on iPhone – Apple Support
says:
"You can change other Safari settings in the Settings app.
From the Home Screen or App Library, go to Settings 
> Safari, then adjust any of the options.
For example:
Search Engine: Choose the search engine to use when you enter search terms in the search field."
If this setting continues to be not working for you on iOS 16.4.1 or 16.4.1(a):
You might want to send feedback to Apple:
Here's the web-form for feedback:
Feedback - Safari - Apple
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Re: Anyone having same problem?
We are all Apple users, like yourself, in the community, so have no access to the feedback, if any, that Apple receives on this topic. Unless a community member with personal experience of the issue answers, likely no-one here would know.
All the best :-)
