This criticism of the way biologists/geneticists/biochemists approach problems is I think quite misguided and misunderstands both limitation of testing things you can't manipulate with your bare hands, and how subtractive (take stuff out, see what happens) methods fit into the larger process of biological inquiry.
First, when you can't manipulate things directly, it becomes more difficult to tweak systems slightly to learn how they work. This is especially true if you start with something made from things you don't understand, and for which you may not even have a complete catalog. The radio paper raises some issues and acknowledges these challenges, but then sort of glosses over them, saying, "it's complicated, but that's just because we don't understand and don't use the right language."
Second, biologists don't just use subtractive methods, they try to use additive approaches to complement and test the hypotheses that come out of "it's broken when I take this bit out." Biochemists also try to reproduce complex behaviors by mixing only the purified components of the system in question. Biochemists and biophysicists also often measure very precisely physical and chemical properties to mathematically define the behavior of biological components.
The point is biology is hard, and we barely know anything about it. We have to start with simple experiments that point us to what's important and then progressively do better defined experiments to figure out how it works in detail. The next step here is: now that we know what proteins are important, let's figure out how they work exactly. We couldn't have gotten there before trying to knock out as many parts as possible.
TL;DR: this isn't the wrong approach, it's the right one, and given the limitations of the system, realistically the only one we have at this stage. It's also only the first step, now we get to try other approaches on what we've learned.