First and foremost, brewing yeast cells do not need to be stirred to remain in suspension, at least not before high krausen is reached. Most brewing yeast strains exhibit what is known as NewFlo flocculation (the genes that are responsible for flocculation are known as FLO genes). NewFlo strains do not flocculate until glucose, mannose, maltose, sucrose, and maltotriose have reached a genetically set level.
At this point, the only positive that stir plates bring to the table when propagating brewing yeast strains is degassing of the medium. However, CO2 build up is not much of a problem when a culture is pitched 12 hours after it is inoculated.
To answer your question, is why would one want to use a stir plate when it basically brings little to the table and risks exposing the cells to continuous shear stress (even a slowly stirred culture undergoes turbulent flow)? Why not just go with an easier and lower cost method that does not unnecessarily waste resources? If stir plates are the answer, why do we not stir our batches of beer? It would be easily to do with a continuous duty motor and a sanitized stainless steel paint stirrer. A batch of beer would more than likely benefit more from continuous stirring than a culture because stirring would keep the cells in suspension long after glucose, mannose, maltose, sucrose, and maltotriose levels have fallen below the levels encoded a yeast strain's genetics.