I'm unsure how this is actually "disproving" the idea that eggs are a natural byproduct of the chickens reproductive cycle.

Obviously ists not a menstruation the same way as in humans, but there is a lot of functional overlap. We're not throwing this kind of "well technically" over similar comparisons of physical functions or body parts that are widely differently constructed in other animals but serve similar functions. Particularly for the purpose of outlining those similar functions.

Sure, its part of our mass-farming cruelty breeding that this natural process has become much less "naturally" torturous for the animal - we didn't give them a cycle, we just perversely supercharged it. Which is an entirely different debate than "can we call eggs 'chicken periods'?" and this vegan site is clearly just here to make that argument much more than actually correct supposed misconceptions.

Which fair, mass poultry farming is a disaster. But pretending it's "inaccurate" to jokingly call them "chicken periods" when we literally use the word eggs both for chicken eggs and human eggs, is being very selectively pedantic solely for ideological reasons.