Plain egg sets quickly
Egg proteins unfold and link up as they heat. With nothing added, they link tightly. The egg turns firm and a little dry.
Sugar slows the setting
Sugar sits between the proteins. They need more heat to link up. The egg stays tender, but the surface browns sooner.
Dashi spreads the proteins out
Adding liquid puts more space between proteins. The set egg is softer and holds broth. Too much liquid and it will not hold a shape.
Cook three small omelettes. Make one plain, one with sugar, and one with dashi. Press each with a spoon and compare.