The problem is not that it's fallible, but that there's no noncircular explanation for why it seems to work. Inductive reasoning doesn't have a rational explanation, only an intuitive one.
I don't think your claim is possible to prove statistically - what does "pure reason" mean? And of course it's "pure reason + empirical observations": even if they're not running reproducible experiments they still see natural phenomena.
It works because humans looked for a method that works, they tried different methods and chose one that works best, actually it gradually evolved over time.
Pure reason is conjecture with little empirical input (and big intuitive or memey input) and little verification of output, i.e. a large deviation from empiricism.
I don't think your claim is possible to prove statistically - what does "pure reason" mean? And of course it's "pure reason + empirical observations": even if they're not running reproducible experiments they still see natural phenomena.