Looks like a pretty good invention. Imagine you have guests coming over, and you need to make lots of sandwiches. Cutting egg slices from an egg cake like this sounds reasonably convenient to me.
It's already a product. It was invented by a Danish company in 1974. Unsurprisingly it's called "langæg" = "long egg"
It's used widely by restaurants and sandwich cafés.
It's not made as a cake though and I'm not sure if OPs recipe would even work, as it'd likely mix before setting.
The way they're made on the factory is by separating the yolks from the egg whites by centrifugation. Basically all the eggs that aren't sold due to cracks or sizes or shell color are tossed into a centrifuge. The white and yolk have different mass, so they separate easily that way.
The white is then slightly cooked while being centrifuged again at somewhat slower speed.The yolk can then be added and cooked fully.
Look again more carefully. The objectionable part isn't actually the egg, it's the granulated sugar coating.
Lots of other folks in this thread are (rightfully) pointing out that hard-boiled eggs can taste fine when baked into things, but all their examples are savory and this one isn't.