Sorry to say that iOS simply won’t allow choosing something other than Safari as the default browser, and Brave (and everyone else apart from Apple) can’t do it. The article you link mentions workarounds from other apps that let you tell them to open things with other browsers, but that’s not the same thing at all. So far, the only way to get what you (and I) want is to jailbreak your iPhone, and I’m not willing to do that.
That said, I’ve used Brave as my main browser on iPhone for … years? It’s in my dock and it’s what I open when I want to use the web. If Siri or another app opens a webpage in Safari and I care about it, I copy the link and switch to Brave. But everything I’ve found says the only way to switch the iOS default browser to Brave (or anything other than Safari) is jailbreaking, and again, that’s a non-starter for me.