No, even some of the double action's of the cap and ball era had conventional grips.(very interesting era) and 7 - 8" barrels in .36 and .44, .45, and even .52cal (brit Harvey).
The Starr, Pettingill, Savage North Figure 8, Adams, and Tranter civil war era double action cap and ball fired revolvers, in the case of the Savage, a gas seal revolver.
And the British made Harvey the only striker fired cap and ball revolver at the time.
Also tacked on a photo of a M&H coil spring fired self cocking mechanism mod fitted to a 1860 colt.
Funny thing Mershon and Hollingsworth also produced a Revolving Cylinder Automatic Rifle, in 1855.
Semi-automatic, but capable of fully-automatic fire, accomplished by winding up a spring using a ratchet.
And lastly a under hammer fired revolver of a different flavor.
The British made J.R Cooper patent underhammer double action pistol, double action revolver fires from the lower chamber of the unfluted cylinder via an underhammer mechanism. At first glance, the barrel appears to be upside down, but it is in fact installed correctly and has an interesting, flip up, tall front sight.
Even the DREYSE needlefire revolver of the 1860's sported a plow handle grip.
What does all the above jee wizz information reveal?
Mid to late 1800's Firearms development in Europe and Americas were pretty much on the same page at the same time.
The saw handle birds head grip on American belt revolver's didnt come about till use of metal cartridges were in common use.
The last pair of photos shows the plow handle single action cap and ball LeMat Revolver vs. The metal cartridge firing saw handle model.
I can only guess it must have been fad?
I have a Ruger bird head grip model .45 and I love how well it handles, why they never csme out with that grip frame earlier is a very good question.