> You just moved it down a layer.
Nope.
> What choice of axioms would lead to desired outcome x?
Yes, that is the part about figuring out consequences.
> Once you've decided on desired outcome x, science can help, but now the philosophical question is: "How should we determine what the desired outcome should be?"
No, that's really a bad question. Or rather, it's first of all an ambiguous question. One can interpret that question as "What method should we use to determine the desired outcome, given a desired meta-outcome" or something of that sort, which is obviously just another question about consequences in disguise, and thus is obviously one that is accessible to empiricism. I'm guessing that's maybe not what you mean, but I just want to make sure we don't chase another equivocation. So, if we ignore that interpretation, what remains is a bad question, for two reasons:
1. It's similar to "intrinsic value", in that "should" without a goal it is directed towards is ill-defined, so you are essentially asking "What do you have to do to reach a goal, absent the goal?", which is a similar nonsensical structure to "What's the judgement of a valuer, absent the valuer?"
2. Even if we suppose that you had an answer that X is what you should do, regardless what your goal is, that answer would be useless. If you are talking to someone who already agrees with you that X is the thing they should do, it's pointless to say to them "you should do X!", because they are already doing X. And if you are talking to someone who does not agree with you, it's equally pointless, because simply stating "you should do X!" has zero potential to convince them of anything, even if that statement happens to be true somehow.
If you want to convince someone that they should do something, you have to show them how it helps them achieve their goals. Just saying that they should do something regardless what their goals are won't get you anywhere. If someone's goal is to kill as many people as possible, telling them "you should not kill other people, and that is the truth" won't have any effect whatsoever. You have to either show them a conflict with some other goal of theirs that they might consider more important, or you have to create that conflict (such as, if their goal is to live in freedom, by threatening to put them in prison while being credible that you can actually make that happen).
> Consensus? Why is consensus better than just going with what zAy0LfpBZLC8mAC believes?
That question doesn't make sense. It's not a matter of whether consensus is better than something else, it's just that consensus is what is, whether it is good or not. Either you get some significant proportion/the majority/everyone to do adopt a certain view, or you don't. Going with what I believe is not an alternative to consensus, it's just one of the many options as to what could potentially become the consensus, and if there is no consensus that you should go with what I believe, it is of no consequence that what I believe is better than what the current consensus happens to be.
You might as well be asking why digestion is better than eating apples. It's just a confused question.
And just to make sure we are not equivocating: This is by definition not about the question whether we should adopt consensus based governance structures, say. Because that is a question about consequences, and thus a question that is accessible to scientific inquiry.
All these questions as to "what is the ultimate goal/value/intention/taste" are simply the wrong question, in that they make an unfalsifiable assumption that makes any answers to those questions useless, while at the same time leading you to ignore the interesting questions that actually have some solid answers, which also largely obviate the need to ask those pointless questions.
Questions such as "How did it happen that humans have a social structure that uses consensus between individuals as a survival strategy?", or "How did it happen that humans have a tendency to help other humans even when there is no immediate benefit to them?", or "How did it happen that humans generally prefer happiness to suffering?", but also "Is it actually true that helping other humans without an obvious immediate benefit indeed does not have any benefit to the person helping?".
Those and many other similar "how" questions have quite a body of knowledge to answer them, with loads of evidence to back up those answers, and some very interesting and often highly unintuitive insights, and understanding both the questions and the answers is way more intellectually satisfying than making up some "ultimate invisible valuer" in order to prop up nonsensical concepts like "intrinsic value" or "ultimate should".