I don’t think installing an app and rooting a device are fair comparisons
Unfortunately, installing an App doesn't let you change the system-wide search on iOS (or Safari browser), so rooting the device would be the only real way. My intended point is that if you're a consumer trying to change your search engine to a new option on an iPhone, there is no way to do it.
I think you might as well be asking regular consumers to root their device so they can use whatever Apps from outside the App Store, or whatever search engine they want.
Also, even for a technical user, there is simply no way on an iPhone to change to a new search engine not already on a tiny list, and from talking with hundreds of consumers, I have not talked with a single non-technical person who could work out themselves how to change their Safari search engine to even one of the 5 limited choices, let alone a new option.