In many ways this is just an extension of his 4 quadrants of conformism essay, so I'll argue with it using ideas from that essay. I would argue that people vastly over estimate their degree of independent mindedness because although it's easy to be independent minded in one field, especially if it's one you have been studying for a while, it's incredibly easy to fall victim to common sheep-like mentalities about any other topic you're not an expert in. Just ask non-economists what they think of the economy, all of their answers, if not outright incorrect, will be canned answers stolen directly from whatever media outlet they use.
This is important to note because literally everyone does it. I don't actually know of a single person that has not done this, and I personally have done it so many times I question my degree of independence and free will all the time. So what can we say about this? Is no one independent minded then?