Both triangular and circular handles have existed since gymnastic rings were invented in the early 19th century.1 The circular style quickly became the sole variety used in gymnastics.
Speaking from experience, I would put this to the following reasons:
- Round rings provide room for a false grip, a technique in which the wrists are hooked over the rings, as the first step in a muscle-up. Triangular handles would probably need to be at least 50% larger than the ones you linked in order to make this possible.
- Round rings can rotate, allowing the user's grip to settle into a stable position. This is especially important when taking the false grip, as the point on the rings where the athlete's weight hangs shifts from the fingers to the wrists as the athlete rolls their wrists up onto the ring. Performing this on a triangular grip would result in the athlete's weight no longer being centred on the grip, and the triangle hanging at an angle.
- Less interference with the body: In many techniques, the rings will come into contact with the athlete's chest. Triangular grips have points which could catch on or dig into the athlete's body. Circles do not.
- Greater stability when positioned above the rings: In positions where the athlete holds their body above the rings, such as the planche, the top half of the rings can be used to press into the forearms for stability. A triangular grip would press into the wrist instead, possible by the point, which would be painful and less stable.
As for why triangular grips exist today, these are generally products used for suspension training, in which trainees perform exercises such as rows and pushups, with their feet still on the ground. The trainees do not need to swing or move between positions while fully suspended. In these cases, the trainee does not need the versatility of the rings, and the flat bar of triangular grips is likely a more familiar grip to them compared to rings. Triangular grips can also be made smaller, making them cheaper and advantageous for transport.