Y = chance there's a real correlation

The y-axis shows the effect size (the odds ratio in this sample), which is different from statistical significance (the probability that the relationship observed in this sample would occur purely by chance). This passage:

you would expect that the trials with small samples would have a wide scattering of results (sometimes they would look wildly significant and sometimes they would look wildly insignficant)

should also be written in terms of effect sizes, rather than statistical significance.