/u/Brigantus presents a great response to this question here, with some additional details here and here

You may also be interested in /u/Tiako discussing "Classical" era Stone Age peoples.

To summarize the above posts:

The "Three Age" system is relic from earlier days of anthropology still in use because of convention and habit. They have been substantially further subdivided and correlated to absolute dates. They no longer represent "stages of advancement," since such a concept is fundamentally useless as a chronological system.

/u/CommodoreCoCo has already linked to a longer post I wrote on the three age system, but I'll expand on it a little bit to address your specific question of why we don't use agricultural technology instead.

The three age system originated first and foremost as a way of dating archaeological finds. The observation that stone tools are more primitive than metal ones, and that bronze is more primitive than iron, goes back centuries, and is pretty obvious if you think about how those respective materials are made. But our modern scheme of Stone, Bronze and Iron Ages goes back to the work of Danish antiquarian Christian Jürgensen Thomsen in the early 1800s, who was able to demonstrate that tools made from each material were associated with a distinct period of time with its own characteristic set of finds. So from the beginning the importance of tool making technology wasn't that it was thought to be especially socially important to people in the past, but that it was very useful for archaeologists trying to date things. In the two centuries since Thomsen's work our dating methods have been considerably refined, of course; first through more detailed studies of different artefact types (where again the emphasis on finds that happen to be useful for dating, regardless of whether or not they're important in other ways), then with the advent scientific absolute dating methods. But because these new methods were developed within the broad framework Thomsen came up with, and because the three age system is thoroughly intertwined with the way prehistory is taught, most archaeologists still use the conventional system of Stone, Bronze and Iron Ages as a useful shorthand.

It wasn't until much later, with the work of V. Gordon Childe in the 1920s, that prehistorians began to appreciate how significant a development the origin of agriculture was. By that time the three age system was already entrenched and was the basis for much finer systems of periodization (Arthur Evans had already divided the Bronze Age of Crete into 14 sub-eras, for example). Moreover, Childe didn't really even know when or where agriculture came from. He guessed the Near East, but he wasn't proved right until the 1950s, and dating the transition relied on already established chronological schema derived from the three age system. By that point agricultural technology was never going to unseat tool-making technology as the basis for an era system. For one, it's not a very good chronological marker. With tools and pottery, you can pick up a single piece and with a decently trained eye date a site to within a few millennia. On the other hand, ascertaining whether agriculture was practised on a prehistoric site involves the careful recovery and specialist analysis of charred seeds, charcoal and animal remains. And all that would tell you is whether the site is in the "Agricultural Age" or the "Pre-Agricultural Age". Once agriculture developed, variations in how it was practised had a lot more to do with local geographic and ecological conditions than innovations over time.

More pertinently, by the time we had a good grasp of the chronology of early farming the radiocarbon revolution was well underway. Archaeologists were much more interested in finally finding out how old their sites really were than coming up with a new system of relative periodization that was slightly more socially relevant.