I think reproducibility is a problem, but mostly for unsexy, non-statistical reasons: overstating conclusions, omitting details, fudging data, etc. Ioannidis discusses these in his section on "bias", but that part of his model is completely tautological and does not explain anything. Literally, let u be the probability that the authors are lying; then, as u increases, be astounded that the false publication rate increases!
The phenomenon he actually investigates: if there are many independent research teams, each with a small sample of data, and a result is published if and only if it is positive, and if legitimate positive results are rare - then there will be many published false positives. At a high level, this is known as the multiple testing problem, and has been known for decades. (That's why this paper is published in PLOS Medicine, not a statistical venue.) His point is that control of false discovery may be done within individual studies, but is not done at the meta-level of multiple studies. The novel contributions are the quantitative estimates obtained by carrying out the napkin math.
Unfortunately, these aren’t good enough to convince me that multiple testing (rather than the "unsexy" stuff) is to blame for the replication crisis. Are there really that many teams working on the exact same utterly hopeless study? His quantitative estimates depend critically on the prior probability that a study
should yield a positive - what he calls R/(R+1). This value is unknown, and there is no attempt to estimate them from data. He just uses values that he thinks sound reasonable (Table 4). But I wouldn't trust his intuition: his example value of R=10^{-4}, in a GWAS, is misguided. He considers every comparison
within a single study as a study in its own right. In reality, one would employ a statistical method which accounts for the multiple testing. In other words, he is repeating the same intra-study/inter-study mistake that motivates the paper! (As an aside, I am dismayed by the equality R = R/(R+1) = 10^{-4}).
In machine learning, there are concerns about reproducibility and multiple testing. (Conducting and submitting lots of junk studies is similar to trying lots of predictors on a benchmark, hoping one does well by chance). Well,
empirically, multiple testing doesn't seem to be as big of a deal as we thought. It is crucial to look at actual data.
Oh well. Ioannidis lets his toy model drag him into baseless conclusions. For example, consider his claim that "the hotter a scientific field (with more scientific teams involved), the less likely the research findings are to be true." Conventional understanding is that more scrutiny, bigger datasets, and less stress about funding all lead to better research. But no, the toy model (which doesn’t account for any of those phenomena) says otherwise!
Catchy titles are admissible in sleepy technical fields, like the one I work in, for authors to drum up general interest in their work. This paper does not qualify. The title is absurdly aggressive for these brief musings. "Multiple testing in scientific publication" would have been reasonable. Of course, no one here would have cited that. Because this paper brings popular attention to reproducibility and p-values, lots of statisticians don't scrutinize it. I'm not that kind.