"If I have seen farther, it is because I stood on the shoulders of giants." My wife found and bought a Chuck-It a couple of years ago; we use it for the dogs. She can chuck a tennis ball as far as I can sling one. And Willeke posted the picture of her Chuck-It, I just copied the link.
Your comparison of a Chuck-It to a
jai alai
basket (xistera) is interesting. So is the Wikipedia article on the sport. Those guys can get the ball up to 302 km (187 miles) per hour?!? We need to get those figures to the jokers that claimed that nobody can sling a rock faster than 60 mph (96 km/hr). A sling can put a rock into at least as high an orbit as a xistera can!
As AussieSlinger noted, "stretchy" is mostly undesirable, but a bit of stretch can be nice. On the one hand, energy spent stretching the cords, is not available to make the rock/ball/whatever fly. On the other hand, stretching the cords is better than stretching your muscles/ligaments/joints. I think the basic rule-of-thumb is: if your sling feels stretchy to you, use a less stretchy material.
Of the materials with which I am familiar, nylon "paracord" has about 9% stretch: if I pull a length of cord just enough to straighten it, and measure its length, and then really haul on it, it becomes almost a tenth longer, and it returns to original length when I release the tension. Most of my slings use nylon cords, and they work fine.
Dacron cord of similar construction (3-strand kern or core, braided mantle) has about 2% stretch. I have a couple of slings with Dacron cords.
50-pound SpiderWire fishing line (braided Spectra [ultra-long-chain polyethylene]) has about
1/
4% stretch; my favorite
"performance" sling
has semi-home-made cords with 7 kern strands of SpiderWire, threaded up the middle of some braided nylon mason's line (round braid, no core). But I don't use this sling too often because my finger gets sore (all the pull gets transferred straight to the finger I have the loop on).
I do not know how stretchy silk is, but you can measure it the same way I determined nylon's stretchiness. I would be interested to know how much it stretches, and how your cords are made (braided?).
As for what you sling, yes, tennis balls can be frustrating, especially if you are trying for distance. They are too light and too fuzzy to punch through the air like a rock can. But for short ranges and accuracy, they are good. Having a dog to retrieve the ball also helps. I've figured out how to roll the ball into the pouch with my foot, so I don't get slobbery hands...