How about asking what you should do? By refusing to ask that question you haven't explained morality, you've merely stuffed it under the rug. Since the should question actually is important (and pressing) you'll find yourself having to sneak in connotations to answer it.
For example, you wrote:
There are questions about how certain outcomes make you feel. There are questions about how people actually act. There are questions about what actions would lead to the world being a "better place" (however you define it).
It's possible to substitute add should-like connotations to any of the above questions and end up with a (generally bad) theory of morality.
What determines whether or not you "should" do something?
My thoughts are that "should requires an axiom". You could say "you shouldn't kill people... if you don't want people to suffer". Or "you should kill people... if you want to go to jail".
In practice, I think people have similar ideas about how outcomes make them feel. Outcome X feels just. Outcome Y feels unjust etc.
When people use the word "should", I think they're implicitly saying "should... in order to achieve the outcomes that me/society feel are just".
I was stunned to read the accounts quoted below. They're claiming that the notion of morality - in the sense of there being a special category of things that you should or should not do for the sake of the things themselves being inherently right or wrong - might not only be a recent invention, but also an incoherent one. Even when I had read debates about e.g. moral realism, I had always understood even the moral irrealists as acknowledging that there are genuine moral attitudes that are fundamentally ingrained in people. But I hadn't ran into a position claiming that it was actually possible for whole cultures to simply not have a concept of morality in the first place.
I'm amazed that I haven't heard these claims discussed more. If they're accurate, then they seem to me to provide a strong argument for both deontology and consequentialism - at least as they're usually understood here - to be not even wrong. Just rationalizations of concepts that got their origin from Judeo-Christian laws and which people held onto because they didn't know of any other way of thinking.