Why couldn't you, say, mix the egg with the other ingredients while everything is still cold, and then slowly raise the temperature of the whole mixture?
You could (and when cooking by sous vide, that's exactly what you do).
It's the "slowly raise the temperature" that gets you.
Sous vide guarantees that not one bit of the mixture will ever get above the set temperature.
But when the mixture is heated in a pot, the bottom surface of the pot will almost always get hotter than the maximum desired temperature of the food, so any mixture that stays in contact with the bottom of the pot for more than a small fraction of a second will be raised to too high a temperature.
The egg proteins will instantly denature, and you'll end up with lumps of scrambled egg.
Some people can get good results, but it requires very gentle heating, continuous stirring and scraping, and a lot of practice.
For us mere mortals, it's so much easier to heat the mixture without the egg, and then when it is at the appropriate temperature quickly stir in the egg and it won't get hot enough to congeal.
Mixing the egg with some of the liquid first makes it thinner and easier to combine with the rest of the liquid.
The key principle in all of this is how protein cooks.
All cooking is a combination of temperature and time, and some foods can be cooked quite well either at low temperatures for a long time or at higher temperatures for a shorter time.
For protein, temperature is by far the dominant factor.
It can be cooked for hours at too low a temperature and still be raw, or can be cooked for one second at too high a temperature and be overcooked.
This is very evident with eggs.
Consider breaking an egg into a hot frying pan.
As soon as the egg hits the oil, some of it instantly turns white (it's already cooked), because the part of the egg that was immediately in contact with the heat was raised above its cooking temperature.
The rest of the egg remains liquid for a while because it is more isolated from the heat and is still below that temperature.
The whole point of tempering is to ensure that none of the egg reaches its critical temperature before it has been thoroughly incorporated into the other ingredients.