I like Polanyi, but positivism is still very much alive. I mentioned in my post various failed attempts to surpass it, in which I would include Pragmatism, continuations of Marxism, the critical school (excluding some late works by Horkheimer), and postmodernism, all of which retain positivism’s negations.
EDIT: to supply some evidence, in Book III of Science, Faith, and Society, Polanyi argues that unchecked Positivism would result in a society not fundamentally dissimilar from the Soviet society which had prompted him to write the book in the first place. I encourage you to read it and compare it against your experience, asking whether Polanyi secured victory, or whether crucial aspects of his thought went unheeded.
1. Polanyi's criticism: Human observers are not objective. They start with their own biases. That means that any observations of events cannot be 100% trusted as a source of truth.
2. Francis Schaeffer's criticism: Within the structure of positivism, there is no way to know that what you observe is actually data. It has no basis for saying that it should be data.
3. Greg Koukl's criticism: Since positivism is a set of statements in epistemology, and not either direct observation or true by pure logic, positivism says that you can't know that positivism is true. It is self-inconsistent.
Positivism "alive" only by inertia. The position doesn't need to be "surpassed" by something, it only needs to be refuted in a way that can't really be answered - and it has been.
Anyway, none of that was my main point. My main point was that it's not positivism that is driving the closing of minds and narrowing of discourse on campuses.
I don't think this is right at all. First, positivism is dead. Michael Polanyi killed it (I forget the name of the book at the moment).
Second, when people complain about "the incredible narrowing of permitted opinions on campuses", they don't mean that only positivist voices are allowed. And those doing the narrowing aren't doing so from a positivist position at all. They're doin so from a position that regards positivism as an attempt by a power structure to assert dominance (and probably racist besides).