Okay, here's my reaction on having consistently having eggs on-hand for the last 2 months (never having done so before):
The biggest plus about them is having something hot/warm to eat in a very short time. They taste fine/unremarkable, better with cheese sprinkled in, but it's about the short cook time.
I'm a microwave guy. Browning meat is about as much time as I want to spend 'cooking' something. Eggs cook fast, and that's great.
But what I did not experience at all was any inkling to eat them in the morning before work. Not at all. Now sure, the above statement about time comes into play here for people who are in the habit of eating them for breakfast, but again, that's just tradition in their lives.
I enjoyed eating 3-5 scrambled eggs as part of a meal for lunch or dinner, paired with some kind of meat. Perfectly fine meal. Quick. I get that. Cool.
So I still have to say, outside of doing it because you've always done it that way, there's no reason for eggs to be a breakfast food other than the egg producers wanting that niche in the food industry.
So yeah, that's it. Will I continue buying eggs? Probably. Have I been missing out on something great? No. But they're fine. Am I going to start experimenting with them? No.
And about 90-95% of the time I eat them, it'll be after noon.
IMHO with the parts I bolded, you're missing one of the points...
One of the advantages of eggs for breakfast is that they're a fast, nutrient and protein-dense, easy to prepare, meal. Throw a little cheese and some onions/tomatoes into a scramble, and it's basically a perfect food to start your day. And it's really, really, easy.
The point of most "breakfast foods" is that they're fast and easy. Bagel toasted with cream cheese. Simple toast (or avocado toast if you're fancy).
The idea for working people is to try to cram as much healthfulness into a meal that they're not going to take 30-60 minutes to prepare. There's no reason you can't have burgers or pizza for breakfast. But they're a HELL of a lot more work than a basic scramble, or a pre-made bread or croissant product that you just warm in a microwave or a toaster.
I think pizza is a great breakfast food, honestly. Carbs, dairy, protein, a tiny little bit of vegetables (tomato). The difference is that while I'll re-heat pizza leftovers for breakfast, I ain't got the time nor energy before a work day to cook a pizza from scratch. And I'm certainly not firing up the grill to cook a burger for breakfast.
That's not the case with eggs. A quick scramble, or eating previously hard boiled eggs, or a fried egg, are all things that take VERY little effort but give you the calories and nutrients you need to prepare yourself for the day.
For "breakfast foods", the short time and ease of preparation is a huge part of the reason why they're "breakfast foods".
