I know at least one patent author for whom the system worked exactly as designed. That being my uncle. Imagine the literal bearded tinkerer in his garage inventing a (much) better sprinkler head. What's the next step? Without the patent system, I guess raise money, build a factory and establish yourself in the market? With the patent system, he sold it to one of the largest manufacturers and today AFAIK about a third of sprinkler heads sold in the US are his invention. He was paid handsomely for the decade plus of tinkering, the manufacturer got a great invention without spending a decade on R&D. Win all around.

Notably this was a physical object and not maths.