Which is neither here, nor there, and has little to do with the question, which is based on the paradox that you need a chicken egg to produce a chicken, and a chicken to produce a chicken egg.
It's not about whether some other random eggs existed before, but about when the cycle of chicken and chicken egg came to be, and how the vicious circle was resolved.
Edit: Even if you specify, that it has to be a "chicken-egg" you just have to decide, whether an animal producing a chicken-egg is automatically a chicken. I would argue no, because you could then prove by induction that everything before has been a chicken.
And I just explained that that alone is neither here, nor there, as the paradox is about the chicken egg, specifically, not whether eggs existed in general.
>At some point some of those egg-hatching animals had mutated enough to be now called the first chicken
Yes, but that's a different answer, which wasn't available when the chicken/egg question was posited - and it still points to a random (evolutionary) and fuzzy (when? how?) process.
We should come up with a better alternative paradoxes ...
Yes, but those initially positing the ancient chicken/egg paradox haven't discovered the theory of evolution yet, and for those later that still use it, the boundaries (when it "had the right gene combination for us to call it a chicken") are fuzzy. Those are the things the paradox aimed/aims to showcase.