I'm not sure what you mean by "putting Born's rule in by hand". We're talking about interpretations here. Born's rule is just an empirical fact. Do you mean that I need to explain why the probabilities are the square of the amplitude? That's kind of like asking me to explain why the speed of light has the value that it has. It doesn't have an explanation. It's just part of the Way Things Are.
But I can make the following heuristic arguments:
1. Outcomes are probabilistic because this is the only way that classical behavior can emerge from the wave function, and without classical behavior you can't copy information, and without copying information you can't have discussions like the one we're having. It's an anthropomorphic argument (actually it's a info-pomomorphic argument :-)
2. The probabilities are the square of the amplitude because that is how you get a useful mathematical model of reality. It is simply an empirical fact that destructive interference happens, so if you want to build a mathematical model of that you need something that can take on negative values. You can't have negative probabilities in classical reality, so the underlying reality must be something other than classical probabilities. The square root of the probability is just the simplest mathematical model that explains the observations.
Maybe this will help: the technical paper that my position is based on is here: