I see what you’re saying, and I agree with it, but the question isn’t asking “Which egg was the first chicken egg?”, it’s asking “Did the egg come before the chicken?” Determining the exact point is a way of answering the question, but is a lot of work that isn’t strictly necessary to do so.
We can use the Theorem because we don’t care when that point actually was, the question doesn’t ask that. We just need to prove that there was such a point, and the Theorem does that.
To use that text as an analogy, we don’t care which is the first purple or blue word, we just know there is one because the gradient starts from red, passes through purple, and ends up blue, so it must have a first purple word and a first blue word.