Oddly, everyone seems to get a different model:
What are these used for?
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It's always difficult to say exactly why SpaceX does things, as they are so secretive. However, the practice has historically been done in spacesuits to prevent ballooning when the suit is pressurized. In particular:
Without such a restraint, the shoulders of the suit under pressure tend to spread apart laterally. This severely limits shoulder mobility; the astronauts cannot move their arms properly.
The usual solution is a bar or strap across the chest, which limits lateral ballooning. Here is an experimental World War II flight suit with such a bar:
B. F. Goodrich developed the XH-1 suit, part of a series of increasingly better suits that were ultimately judged the best of the Word War II efforts. Bellows at the elbows and knees provided improved mobility. Note the metal rod stretching between the shoulder rings, an attempt to control side-wise ballooning. This would become a feature on most of the subsequent suits designed by Russell Colley.
Here you can see a strap across the Russian Sokol intravehicular spacesuit, also used for the nametag:
And a similar white strap across the chest of the Mercury spacesuit:
The SpaceX quickdraws would be adequate for this purpose.
Without a restraint, the fabric at the shoulders rises up, forcing the neck ring and helmet to also rise. The astronaut's mouth and nose are now down in the neck ring, with only the eyes and top of the head now visible in the helmet faceplate.
The solution is a "helmet holddown" cable or webbing running from where the neck meets the shoulder, to the groin. In the Sokol suit (above), this is hidden inside the suit, and runs along the entry zippers. In the Mercury suit (also above), you can see silver cables in a V shape to the middle of the chest, where they meet a white strap that runs to the groin.
The SpaceX quickdraws should help somewhat, although ideally they should attach at the neckring, not at the corners of the shoulders.
Apollo spacesuits had restraint cables that were hidden inside the suit.
American (EMU) and Russian (Orlan) EVA suits used with the Shuttle and ISS have a hard upper torso, which is rigid enough to prevent ballooning.
Gemini and the intravehicular Shuttle suits (ejection seat suit, LES, ACES) use an ingenious layer of nylon mesh called Link-Net to prevent ballooning:
The restraint layer featured a major break-through with the use of a new “distorted-angle fabric” called Link-Net, which was used to control ballooning and enhance range of motion. This eliminated the need for the tomato-worm bellows at the limb joints used by ILC and most of the World War II suits. Link-Net is a series of parallel cords that loop each other at frequent intervals. The loops are interlocked but not connected so that the cords can slide over each other and feed from one section of the suit to another. [...] The enormous advantages offered by the Link-Net fabric were initially hard to fully grasp. The fabric allowed David Clark Company to design a suit that provided reasonable mobility without resorting to complicated mechanical or metal joints, saving considerable weight and providing much better comfort.
Dressing for Altitude, pp. 248-251