But if the proposition was "belief in a cortex-wide neural code" and the x axis was "years working in neuroscience", would the correlation still be uninteresting?

If there are thousands of people working in neuroscience and you present a poll of 16 of them which shows correlation between the results and how long they've been working in the field, and you leave out how you selected them and why they each think what they do and how they might respond to one another, then I wouldn't assign much credence to the results.

Obviously I'm suggesting that some of the dismissal of the correlation might be due to bias

Or the correlation might be to the pollster's bias in choosing respondents. Or (most likely) it might be accidental due to underpowered analysis.

To me, if belief in a proposition in field X varies inversely with experience in field X... well all else equal that's grounds to give the proposition a bit more scrutiny.

To be clear, I'm saying that this study is far too underpowered, underspecified and under-explained to cause me to believe that the "belief in a proposition in field X varies inversely with experience ". If I believed that, I would come to the same conclusion as you do.