I think there is some truth to the "coolness" factor of the tri-knob handles, but I would characterize it more as a "trendy" factor, or if you prefer, something where people do what others do because it is available and the others are successful. Kind of like baseball players who cork bats or hollow out the tops. I love a good argument (I tell my wife it is a discussion), so here are some more thoughts.
Here is a Shakespeare 1740 HE with a very large Jeffrey tri-handle. The reel is freespool. It came from an active ACA caster. I can see the value of this handle, which is not in the freespool cast, but in the ease of retrieval using the handles which I tested and believe to be superior in feel to the 2-knobed factory version.


With the tiny 3 handle knobs such as the ones we see on the older Meek reels, I think the value is in limiting the inertia on the cast, since most of those reels are not freespool. I can see no value in the retrieval on the smaller versions.
It has been pointed out frequently that in tournament casting, you want the spool to be low inertia, that is, you don't want the spool to outrun the line. On most of the Kentucky reels without aluminum or wood spools, they spin fast and forever. So adding an aluminum spool and a tiny handle would have resulted in a more desirable state where the weight and leader pulled the line off the spool with less (or no) overrun.
So my conclusion is, tiny handle, big cast, except on freespool reels, where it is big handle, fast retrieve. But just my thoughts, no expertise here.