almost looks like some sort of nearly random process except that the probability of a new discovery or invention increases as more discoveries occur.

And as population increases that would tend to increase the rate of discovery or invention as well. This is basically Julian Simon's argument in The Great Breakthrough and Its Causes, that gradually increasing population hit a point where the rates of discovery and invention suddenly started increasing rapidly (and population then started increasing even more rapidly), resulting in the Renaissance and ultimately in the Industrial Revolution. He gives some thought and argument as to why they didn't happen earlier in India or China, but I think the specific arguments a bit iffy.