Common descent means you can only behave as you do due to historical accidents, which may not hold for much longer as technology improves.

There is an unbroken series running from any human, to any other animal (like a chicken), where each two successive individuals are mother and child. It is merely historical happenstance that most of the individuals, except those at the two ends, are dead: and one that will no longer hold in the future if we defeat death.

If you force your moral theory to evaluate all the individuals along that line - who, again, are not hypotheticals, counterfactuals or future possibilities, but actual historical deceased individuals - then it might say one of two things:

1. At some point along the line, the mother has moral value, and the child and its descendants do not (or vice versa). Or, at a special point along the line, one daughter has moral value, and her sister does not. That seems arbitrary and absurd.
2. Moral value gradually changes along the line. People who hold this view differ on whether it ever reaches effectively zero value (in the case of chickens, at least). In this case, you necessarily admit that pre-human individuals (like your ancestor of a few million years ago) have distinctly less-than-human moral value. And you (almost) necessarily admit that, if you extend the line beyond humans into some hypothetical futures, individuals might come to exist who have greater moral value than anyone living today.

The second option flies in the face of your position that:

I think we end up with a much better society if we treat all humans as morally equal

Historically, this merely begs the question: it shifts the argument onto "who is human?" Are blacks? Are women? Are children? Are babies? Are fetuses? Are brain-dead patients? Are mentally or physically crippled individuals? Are factors like (anti-)contraceptives influencing the possibility of future children? Ad infinitum.

So the real value of this statement is merely in the implication that we agree to treat clear non-humans, like chickens, as having no moral value at all. Otherwise we would be forced to admit that some chickens may have more moral value than some proto- or pre-humans.