But that's irrelevant. The argument isn't about where eggs come from, and simply laying eggs doesn't make something a chicken.
If we define "chicken" as the biological creatures we know today that has a certain genetic makeup, then at some point in their evolutionary history there would be the first such creature that has this makeup being hatched from eggs made by creatures that don't. Just as, at some point in the past, there was the first genetic modern human born from pre-human parents.
Hence, the egg must have come first. This is only an unsolvable problem if you hold that there have always been chickens that never evolved from earlier creatures, i.e. the Creationist model.
What you describe is more like Deism or Theism, where you believe in some sort of higher order intelligence behind the universe, but it's not incompatible with science and rationality and observation. You can't believe in both Creationism and evolution because Creationism is an explicit denial of evolution, and of anything but a literal reading of Genesis. But you're in good company, as you just described many/most of America's Founding Fathers, who today's Evangelicals would have lynched as heretics if they bothered to read their actual words rather than treat them as cartoon mascots for whatever they want to imagine America stands for at the moment.
And why do all the creatures (you claim were created out of thin air) as if they had common ancestors? Both physiologically as well as genetically?
To quote Dobzhansky: Nothing in Biology Makes Sense Except in the Light of Evolution.
Why Evolution is True by Jerry Coyne might help you resolving that last part of the puzzle.
My vision is one I grew after reading several texts of several sources and reflecting about them, not only traditional religious texts but more esoteric and occult texts about it. It's very interesting to see that they mostly complement one another. If you get an esoteric hermetic text or something about how creation occurred you see that it doesn't negate the Bible, and it doesn't negate Science at all. It just goes deeper and deeper, mostly using symbolism.
>at what point does it stop being a chicken, i.e. its parent.
When it doesn't meet the genetic definition of a chicken, however you choose to define that. Once you have a definition, there will be a point in evolutionary history where such a creature arises. Anything before that is not a chicken, so therefore
>is that itself the egg or the chicken-maker
Once again, the answer is "the egg". A pre-chicken creature produced an egg that hatched the first chicken. Problem neatly solved.
I thought I was on the right road with Theseus, but maybe not.
I had the opposite problem, with parochial school teachers well versed in shifting back and forth between "it was a metaphor" when evidence contradicted it, to biblical literalism when there wasn't clear contrary evidence. So it was right when right, and meant to show us what is right when it was false.
I thought C&E was a philosophical question, like the ship of Theseus, it can't have a reductive answer because it represents ideals rather than logic.
To be devil's advocate for your stance, the '... an egg that was a mutation from non-chicken parents' were themselves derived first from an egg, which could be said to be the origin of the chicken, but itself laid by antecedent.... So eventually it must go back to the first entity that laid the first egg, and whenceforth they, etc...hence a philosophical perspective.