Oh I'm sure there are people who are reading this that think we are dead serious and have some sort of scientific data to back it all up (and that we have probably made our reservations at the local looney bin).. maybe we need a giant disclaimer stating THIS IS JUST FOR FUN!..I'll play another round. I'm kind of relunctant because I expect someone to read this and misunderstand it is just a hypothetical game. Somebody is going to think we are serious about this. I worked with people before I retired. It's amazing how easy it is for a false rumor to get started.
Purely hypothetical assuming that you could sex them by egg shape (which I still don't accept) and that egg shape is inherited and sex linked. I believe your question is wouldn't these become extinct if they developed a line that only laid eggs that produced females.
To breed this chicken that only lays "female" eggs, they would have to originally breed a rooster to a hen. You would have one line of chickens that breeds the roosters and one line that breeds the hens used to make this "super-hen", much like they do now with commercial broilers and egg layers. The only way this could be sex-linked would be for the "male" trait to be dominant and the rooster to be pure for the "female-only" gene and the hens to have the "male" gene.
Based on what I've seen in my flock and what has happened in industry, I'd venture to say this sex-linked sexing gene does not exist, whether it is tied to egg shape or not, so this is hypothetic only.
Kovalines, mine are a mixed "mutt" flock. I'll be hatching some purebreds I'm getting from someone else. Those will be randon shapes, but those will be easy to tell apart from my mutts.. I'll also throw some of my mutt eggs in there. All these will be clearly pointy, not sort of and not maybe but clearly pointy, just to see what happens. But mine are not set up for any type of sexlink sexing and I don't vent-sex. Not only do I not have that skill but I wouldn't trust my old eyes anymore for that fine work. It will be months before I know the sex of those chicks.

think even if I had 100 hatches that all came out like my last one I would STILL doubt the whole egg shape sexing thing.. common sense still tells me this is a fluke unless it worked 100% of the time for 100% of all people who tried it
but ya have to admit that it's fun and we are all getting a good chuckle from it