First, crack and pop are really terms that mean something different to everyone. (
) (Crack is discuss at 5:42 in the video) I see crack as the shotgun effect (extreme articulation) that is greatly influenced by head tension (high tension) and the snare wires (perhaps medium tension). Most of the piccolo snares I have seen are metal (steel, brass etc) because they can take the high pressure of the heads. They also have less sustain since they are very shallow (less than 5 inches depth). Finally, many piccolo snares are 13 inch diameter because this allows the high pitch piccolo crack with less tension. Pop I have seen defined as less crack and more body. (less shotgun effect). I have the Pearl 13 inch piccolo snare and it has a very wide tuning range. I have used it for years as my back up to my main snare and sounds great tuned low.
As for the effect of the shell, it is likely way less than 10%. (IMHO) I have seen it written that the heads make up 80% (or more) of the sound. (That is a debatable number and I have seen it stated as higher but from experience it is basis for the majority of the drum sound - it provides the foundation for tuning to your desired sound). That means the rest of the sound is created by the rims, snare wires, other hardware, bearing edge, thickness of the shells, number of plys, type of wood (hard vs soft), diameter, depth, number of vents, tuning etc. The difference between different hard woods is very small.
(
https://www.soundonsound.com/techniques/engineers-guide-tuning-and-damping-drums#:~:text=Apparently, it has been 'scientifically proven' that drum,of vital importance, and is worth special attention )
If you are looking to retain some of the characteristics of a piccolo snare without having a piccolo snare, then I suggest to also look at 13 inch snare drums and shallower drums than 6.5....perhaps 5.5 inches or 5 inch depth. They will provide some body with less sustain (some crack with more pop). Also, tuning higher and adjusting the snare wires may help you find the sound you want. I love my T.L. Custom 6.5X13 inch Maple snare drum. (
http://tldrums.com/) It has lots of body but still gives a nice pop.
Edit: I forgot to mention that on cheaper drums the interior is often not sealed. Sealing the interior helps perent the high frequencies from being absorbed by the wood. Sealing the inside can make a big difference on the tone.