LLM didn't write anything in my post; these are all my words and my choices. Conway himself described surreal generation in short like this in ONAG:

> We may say that Cantor was only interested in moving ever rightwards, whereas Dedekind stopped to fill in the gaps, so that R was always empty for Cantor, never empty for Dedekind. It is remarkable that by dropping these restrictions we obtain a theory that is both more general and more easy to work with.

This is precisely the intuition I present to the reader of the article. I am relying on visual aid (concretely, the ordered number line) to imply the machinery explicit in the actual recursive definition. The intended reader of this article is not a mathematician, and I think intuition is vastly more important here.

And I don't think I'm conflating 0 with {0} as you claim. When I say zero is "between nothing and nothing", I mean 0 := {|}. When I say one is "between zero and nothing", I mean 1 := {0|}. When I say 1/2 is "between 0 and 1", I mean 1/2 := {0|1}. And so on. I elide "the simplest number" because I am already going in the order of simplicity. I do not need to explain that alternative spellings like 1/2 = {0.2 | 1} are valid because it is not relevant to establishing the mental model of birthdays.

For the finite cases in my explanation, I do not need to explain that the left and the right parts form sets because I only ever need at most one surreal on either side to define the next generation. I also do not need to state the left/right order condition because it is already visually implied by the picture. For the same reason, I do not need to explicitly quantify over the set of earlier-born surreals, since in these finite cases, if we go birthday by birthday, each next day's surreals are definable via the numbers already constructed by the previous day.

I agree that these finite examples don't spell out how to handle infinitely many bounds at the omega-th day, which is where I believe the illustration embedded below is more helpful. I still think "a gap beyond 0, 1, 2, 3, ... with nothing on the right" is a useful intuition when we get there.

For a more precise but accessible treatment, I think https://www.infinitelymore.xyz/p/surreal-numbers is much clearer than Wikipedia.