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:
Primary end points were atherosclerotic events, including fatal or nonfatal ischemic heart disease, fatal or nonfatal stroke, and peripheral arterial disease.
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 wa...
(For Round 1, see this comment from last year.)
NYT: Studies Link Daily Doses of Aspirin to Reduced Risk of Cancer
The article is worth reading in its entirety, but here's an especially interesting paragraph:
The evidence still isn't perfect, but the purpose of rationality is making good decisions with limited information. I am a healthy 28-year-old and these studies make me even more confident that taking daily low-dose aspirin is the right thing for me to do.
On a related note, if society were more rational, I wouldn't have to be sad reading paragraphs like this one:
Or these ones from A Cheap Drug Is Found to Save Bleeding Victims, published on the same day: