IMO the reason is purely cosmetic. The original Birdshead grips were on Colt Lightenings and Thunderers from the 19th century. (Model 1877's.) These were handy little pocket pistols with complicated mechanisms that broke easy and fixed hard (still do). In fact the only guy I know that can fix one is a shade-tree gunsmith in Tulsa that is rough as a cob and higher than he//. He's also as old as Methusala and maybe just as dead by now, as its been several years since I saw him. At any rate, these were pretty popular guns in the old west. They look almost identical to a SAA only smaller and with the grip. So people probably wanted a modern, more repairable gun, that looked like these.
The full-size versions look like them but are larger and no such gun existed on the frontier. The smaller ones such as Uberti's current small-frame version in 38 Spec. look very, very close to the originals and differ only in mechanism and the fact that the originals were double action and the current offerings are single, of course. I've thought about getting my wife one. They are very pretty little guns and fit small hands well.
There was at least one other gun with Birdshead type grips but the 1877's were probably the most prolific. In fact, there may have been some that pre-dated the 1877's, so maybe "original" is a bad choice of words. The Merwin Hulbert's come immediately to mind. The grips on the Lightenings look more like today's though.
