There are several cooking factors to control simultaneously, and if you try it, it takes hand agility, coordination and plenty of practice.
But now consider how egg used in tamago fits into sushi. I cannot say it is unappetizing because, it can be tasteful and for sugar lovers, satisfy their sweet tooth. But somehow I think of tamago as out-of-place.
Sushi has been food that has been solely dedicated to seafood and overtime, I guess that leftover Mirin, Shoyu and the fresh morning eggs (not to wasted) were combined to make an omelet. Sushi chef’s are frugal, so it is highly possible that left overs were used to satisfy the chef’s early morning appetite and hence an omelet was made.
There’s even an adage among some sushi connoisseurs: you can tell how good the chef by his tamago. But I don’t agree, and I really can’t see the connection between cooked omelet and the fresh sea foods used in sushi. Almost every single element in sushi is based on the seaside, except for the white rice which is the common denominator.