Earlier quoted context omitted.
I am not a physicist, but this seems backwards to me. If anything, it is the Copenhagen interpretation that comes with the largest amount of extra-mathematical baggage while many-worlds interpretation came out of effort to add the least amount of additional assumptions on top of the math.
The price that MWI pays for this minimalism is that it loses some contact with reality, i.e. physical experiments. Basic physical observations like "some measurements are non-deterministic" are quite difficult to explain in the MWI. Reality is inconvenient.
Max Tegmark gives a pretty accessible description of this in his book - Our Mathematical Universe, and Eliezer Yudkowsky does a decent job with a general explanation here (though he avoids talking about MWI): http://lesswrong.com/lw/pd/configurations_and_amplitude/
I think the Copenhagen interpretation is now considered wrong by most physicists. It's strange to me that it was ever really thought of as true - though it makes sense that it would look like what's happening at first.
It's less that MWI is a predictive theory and more that MWI is something that comes out as a result of the theory's prediction (Schrodinger's wave equation that doesn't collapse).