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gwern comments on Daily Low-Dose Aspirin, Round 2 - Less Wrong
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If you look at the Rothwell paper linked above by gwern, there's one Japanese study there - JPAD (Ogawa H, Nakayama M, Morimoto T, et al. Japanese Primary Prevention of Atherosclerosis With Aspirin for Diabetes (JPAD) Trial Investigators. Low-dose aspirin for primary prevention of atherosclerotic events in patients with type 2 diabetes: a randomized controlled trial. JAMA 2008; 300: 2134–41).
Looking at the graphs, JPAD effectively showed zero effect of aspirin. I think this supports your thesis.
But from JPAD, the specific effects that are supposed to show aspirin benefits weren't what this study measured! They were looking at:
This doesn't include anything related to cancer. And to the extent that the study did look at all-cause mortality, it showed exactly what you would expect from a relatively small (n=2539) and short ("median follow-up of 4.37 years") trial where aspirin was yielding benefits: a large but non-statistically-significant effect
I haven't looked at the JPAD paper yet, but that Cochrane study plots JPAD results for cancer endpoints as well as major vascular and extracranial bleeds. In the same graphs I don't see large effects.
It's worth pointing out the general low cancer rates (in the controls as well) in the JPAD study.