That being said, the belief that the scientific method can build up arbitrarily accurate models of any observable phenomena is without foundation, yet some behave as if it is obviously true.

Until we have something like a unified model of physics, there is no reason to believe that statement to be true.

(and in fact, the scientific method has led to results like Gödel’s incompleteness theorem, which makes it perhaps more likely for it to be false)

Maybe numbers and language (especially written) are more important. Oh ya, and fire.

But the scientific method is way up there in important, no argument there.

Often though, we try to (or pretend we are applying this method to achieve unscientific questions (a la Karl Popper's criticisms of Scientism) or try to add up scientific answers in an unscientific way.

When we do that, we are using the (well earned and underappreciated) cultural authority of science in ways that do not promote truth seeking.