I use Vivaldi browser but the iOS version is lacking some defining features compared to Android or desktop versions. Not delved into alternative browsers on iOS much (Vivaldi is what I use on everything what), but from what I tried they're not as polished as safari since it wasn't that long ago that apple allowed you to even set the default browser in iOS.
Definitely monopolist strategy but unfortunately most apple users see the closed ecosystem as a selling point so hard to see it changing
I’m using Qwant browser which uses Qwant search engine. Both the browser and the search engine are actually quite decent, and I think it’s a French project
Kagi has its own extension that redirects search terms to itself, works most of the time. Unfortunately they’re paid and in the US. I did find an extension for safari that should do the same thing for your use case though. https://mshibanami.github.io/redirect-web/