They don't "stop" performing; they start to perform differently. Autolysis rates go up, alcohol tolerance goes down, and the yeast generally perform in a suboptimal fashion with respect to making good beer. They doesn't just keel over and die after 12 generations, or there wouldn't be any yeast left on the planet.

The yeast cells have internal mechanisms for storing energy and raw materials (glycogen and sterols) and they normally go through a cycle of feast and famine, gene mixing and senescence, designed to increase their chance of survival back when they were surviving by floating from squished grape to rotting peach and drying out between meals. However, they're never starved of oxygen in a natural (think fruit orchard or vineyard) environment. They need oxygen to make sterols, the feedstock for the bi-lipid membrane that protects them from the world and allows them to maintain homeostasis.

The way we brew with them now is very unlike their (semi-recent) evolutionary past, although they will readily adapt to constant fermentation and a sugar-rich environment if used in a brewery for decades. When we brew with them we starve them of oxygen, forcing them to ferment instead of respirate, and produce alcohol instead of greater amounts of energy from glucose. This is a survival mechanism we've hijacked. Yeast in their natural state might go through a short period of oxygen starvation (hence the fermentation metabolic pathway in the first place), but nothing like what we stick them with a brewery. We don't give them a chance to completely restore their sterol and glycogen cache before throwing them once more into a sugary fermentor. The problem is that as they adapt to this energy rich but nutrient poor environment (by that I mean wort), you cannot guarantee that their adaptations won't lead to flavor changes relative to the mother strain. The solution is to restart from scratch periodically to make sure you have no flavor drift and you have cells fully stocked with glycogen ready to go gung ho without a pause in beer production.

Getting back to your point, the brewing methods of the past (namely open fermentors) allowed the yeast to remain exposed to oxygen to a much greater degree so they had a healthier sterol repository, and the way brewers "accidentally" inoculated their new brews (example: yeast slurry on a wooden paddle used in a previous brew) mimicked the wet/dry feast/famine natural state of yeast. Hopefully all that made sense!