Mini golf might be a lot older than you thought. The oldest putt putt course in existence was built around 1867 by the Ladies' Putting Club of St. Andrews. It was considered improper at the time for ladies to 'take the club back past their shoulders.' Because of this, they created a course that allowed them to play full games without doing that.
The first mini-golf courses were actually miniature versions of the larger ones. They had greens, fairways, sand traps etc. It wasn't until after the great depression, when these mini golf courses became too expensive, that people began to build courses with whatever obstacles they could find. Tires, rain gutters, barrels and pipes.
The rest was history. The crazy obstacles became commonplace, and even survived efforts by some people who wanted to bring putt putt back to basics, to just a green.