The meta-analysis you cite is moderately convincing, but only moderately. They had enough different analyses such that some would come out significant by pure chance. Aspirin was found to have an effect on 15-year-mortality significant only at the .05 level, and aspirin was found not to have a significant effect 20-year-mortality, so take it with a grain of salt. There was also some discussion in the literature about how it's meta-analyzing studies performed on people with cardiac risk factors but not bleed risk factors, and so the subjects may have been better candidates for aspirin than the general population.

The Wikipedia quote you give is referring to secondary prevention, which means "prevention of a disease happening again in someone who's already had the disease". Everyone agrees aspirin is useful for secondary prevention, but there are a lot of cases where something useful for secondary prevention isn't as good for primary. In primary prevention, aspirin doesn't get anywhere near a tenth reduction in mortality (although it does seem to have a lesser effect).

I would say right now there's enough evidence that people who enjoy self-experimentation are justified in trying low-dose aspirin and probably won't actively hurt themselves (assuming they check whether they're at special risk of bleeds first), but not enough evidence that doctors should be demonized for not telling everyone to do it.