In fact, the difficulty of disentangling the subject from the objective is a fundamental purpose of scientific method. There is no way to know for sure that results are completely objective for the simple logical reason that there are no absolute criteria available to humans, although different groups have declared them in their favor for ages and continue to do so. That's called dogmatism. The proper antidote, especially in science, is skepticism.
Healthy skepticism doesn't mean that everything is equally in doubt all the time. It simply means that the probability 0 (contradiction) and 1 (tautology) are logical and assert nothing substantial about the world. They are rules about using signs correctly.
When a conceptual model is interpreted are representing states of affairs, the test of the model is comparing it with facts. Here the probability lies between 0 and 1. There is also ontological uncertainty, affecting complex systems owing to emergence, for example, and epistemological uncertainty, for example, bounded rationality.
Analytic philosophy, on one hand, and anthropology, sociology and history, on the other, show that so-called human nature is a slippery concept and human knowledge is influenced by both nature and nurture. Philosophy of science reveals that even so-called facts are theory-laden to a great extent in important cases.
The reason I think that Noah is cavalier about his argument is the excluded middle, which in this case, is broad and nuanced. It's not a matter of whether economics is ideological or non-ideological, but rather about specific cases and conditions, and where method is too lax, logic too loose, and subjectivity creeping in.
There may be cases, usually simple, where the probability of conclusions about correspondence of models with facts and events is relatively high, and others where it is relatively low. In some cases, no probability can even be assigned owing to various limitations. One of the chief issues that economics faces is tractability, for example, which can lead to a tendency to over-generalize and to fall into the fallacy of false cause. Many logical fallacies have been identified historically, and now cognitive biases are being discovered.
Tractability is often addressed by assuming "all things being equal,"when all things are never equal in dynamic situations involving complexity, reflexivity, learning, adaptation, and emergence. For example, most conventional economists criticize Marx for missing the effect of technology, while being unaware of what they may be missing themselves.
Where ideology is most likely to enter is in assumptions, both factual and methodological. This also happens to be where there is much controversy. Macroeconomics is political economy, and where policy is involved, ideology is usually not far away.
The study of this is "meta," because it involves not the practice of economics but the study of the foundations of economics, foundations of social science, foundations of science, and foundations or knowledge.
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