When it is apparent that it refers to the egg (or at least I’d assume that most people also interpret it this way), is it fair to use it instead of some other word?
Or would we have to remove the so-called “ambiguity” by not using a pronoun there?
Could the sentence be improved?
Not really, it's fine.
It's worth noting that while the sentence is grammatically ambiguous in the use of the pronoun, its meaning is not ambiguous. The joke works precisely because you first interpret it one way without any difficulty, and then the punchline insists on interpreting it differently. If the sentence was truly ambiguous, then it wouldn't be a joke.
Grammatical ambiguity is a problem if:
- It leads to semantic ambiguity whether permanent (you really can't be sure what is meant) or fleeting (you can figure out what is meant, but there is a jarring moment where you are briefly unsure).
- The two (or more) ambiguous meanings don't amount to the same thing.
- You do not want to introduce this semantic ambiguity for effect, such as dramatic irony.
The straight (not joking) question, "How can you drop an egg on a concrete floor without cracking it?" does not suffer from that. If we remove the grammatical ambiguity with "How can you drop an egg on a concrete floor without cracking that egg?", it just becomes a bit clumsier, without being any clearer in practice.