Why not incorporate eggs at the last moment when making crepes?

Your supposed advantages are not correct.

I believe if eggs are mixed too much, whites can get a rubber-like texture.

No, this is not correct. Are you thinking of gluten? That is the ingredient that gets tough with overmixing. So by that logic, you should be adding the flour last - but the whole point of recipes which are being held for a long time is to allow time for the starch in the flour to hydrate, and with the longer periods (you mention 24 hours), to allow for the building of slight sourdough flavors. So you cannot add flour last either - or you can, but then you don't need to keep the batter around. Anyhow, there is no change in toughness when you add eggs first.

Eggs are fresher when they are in their shell

They may be "fresher" by common sense understanding of the word, but by food safety rules, the two batters (the one with the eggs added before a rest, and the other with eggs after the rest) are equally safe. The food safety is not increased.

On the other side, if your suggestion is followed, there may be some slight disadvantages in handling. They are not so terrible as to make your suggestion unworkable, but together with the lack of advantages, they provide a good reason for recipes preferring early mixing.

A list of some of the disadvantages:

1. Not only is it more convenient to only get out a whisk once, but there are also people who prefer to mix the eggs with the flour first because this is their preferred method against getting lumps,
2. There are the recipes which require you to start with an egg and then adjust the consistency with incremental additions of flour and milk
3. There is the slight effect of possibly less-well beaten egg diffunding better into the mixture during the resting period. Basically, if you want to go to the trouble of doing a resting period, whose purpose it is to get everything to settle together, you'd better add everything before you rest it.

I must say that personally, I actually do add "egg last", but that's more of a side effect of my method. I use a very simple recipe (only eggs, flour and milk) measured by weight. I whisk by hand, and I have noticed that I get the least lumps when I first whisk the flour with the milk. This automatically means that the eggs (separately stirred, without whipping) come in last. I don't rest my batter, so there is no question of adding it after the rest (and if I did rest, I would add it before the rest). So you see, egg last is not impossible, just like Willeke said, it just has no special advantages, and in some cases it comes with disadvantages.

I have started to make pancakes, and we use a crepe recipe, only to notice I had forgotten to add the egg.

I next added the egg and made the pancakes.

It can be done, eggs last, but I see no advantage.