What is it that is difficult to detect in a person and still people care about potential partners having it?

I guess it is how the person will behave in the future, and in exceptional situations. We can predict it based on person's behavior here and now, unless that behavior is faked to confuse our algorithms.

Humans are not automatically strategic and nature is not antropomorphic, but if I tried to translate the nature's concerns for "I want an alpha male", it would be: "I want to be sure my partner will be able to protect me and our children in case of conflict."

This strategy is calibrated for an ancient environment, so it sometimes fails, but often it works; some traits are also useful now, and even the less useful traits still make impression on other people, so they give a social bonus. (For example higher people earn more on average, even if their height is not necessary for their work.)

Of course there is a difference between what our genes "want" and what we want. I guess a typical human female does not rationally evaluate male's capacity of protecting her in combat, but it's more like unconscious evaluation + halo effect. A male unconsciously evaluated as an alpha male seems superior in almost everything; he will seem at the same time stronger, wiser, more witty, nicer, more skilled, spiritually superior, whatever. A conflicting information will be filtered away. ("He beat the shit out of those people, because they looked at him the wrong way, and he felt the need to protect me. He loves me so much! No, he is not agresssive; he may give that impression, but only because you don't really know him. In fact he is very gentle, he has a good heart and wouldn't wish no harm to anyone. He is just a bit different, because he is such a strong personality. Don't judge him, because you don't know him as much as I do! And he did not really murder that guy in 2004, he was just framed by the corrupt police; he explained me everything, because he trusts me. And by the way the dead guy deserved it.")

Anyway, preference is a preference, you cannot explain it away. (Analogically, if a male prefers females with big breasts, you can't change his preference by explaining that bigger breasts are not necessary to feed children.)