Current morality, due to a lack of properly hammered-out psychology interfacing with ethical philosophy, very much does come down to axioms as a necessity from lack of applicably data. We quite simply don't have a "base truth" that we can build morality out of without resorting to a moral axiom. We don't know enough about the human mind to say "This Is What Is Desired" as a general habit of humans as a whole.
There's also the matter that desires, while not worked out precisely enough to found morality on, are biased by sex, so there's a lot to be discussed with regards to how to weigh the conflicts of interest. Further compounded by variances with age and, to a certain degree owing to genetics-based personality characteristics, also possessing varying values.
While the data to get into what things to promote and forbid in particulars for the sake of personal good is absent, what is present, following from wholly material and observable nature, is that humans, as propagating organisms, necessarily have a "pillar" of long-term reproductive success. Not merely single individuals, but the persistence and growth of populations, in that order. This is further reinforced by social mentalities, which further suggest that the group take priority over the individual.
However, as a contrary factor, and where it comes to working out solutions to conflicts of interest where the absent data on human nature is needed, humans don't cope well with a full absence of personal control, though it takes a vast displeasure to instigate outright rejection of authority on a larger scale. Slave rebellions are relatively commonplace throughout history, alongside many other deprived lower classes, despite stability and, in some cases, outright improvement in standing of the overall population. Therefor, there's some necessary degree of personal value.
More bluntly, the most readily apparent basis for a non-axiomatic morality is maximum average happiness, within a social order that will be stable for prolonged periods. The latter having priority before the former, thus the mere desires of the individuals are subservient to the needs of the group in abstract, and as such outlier cases like the "Happiness Monster" that are destructive to the overall group if desires are met are deprived of those group-destructive desires, but once those needs are handled in a sustainable fashion, it turns to the maximizing of desires being met on average.
The exact nature of the social structure, the desires to meet themselves, what desires are destructive to the wider society and thus should not be met, and basically everything about the details are reliant on data beyond my personal knowledge, and I expect beyond the knowledge of the scientific community, even if they could pull their heads out of their assess and accept that the nuclear family is at least the best starting point we have, for just how dominant it's been across the whole planet, and thus practices that impede it should be weighted against by the selection pressures of society.
I'll also admit that this very much fits the "skeleton" of my own idea of ethics, where stability of needs is the first priority, upon which virtually all other things may be sacrificed. I'll take a thousand President Pinochets over a Chairman Mao. The former is still fewer deaths and less economic destruction (Pinochet's Chile was almost perfectly stagnate, economically, neither improving nor decaying in any critical fashion. Mao, meanwhile, managed to fuck up economically to the point of millions of famine deaths when he started with a sizable food surplus). Of course, I'll also similarly take a dozen of modern Communist China over a single Wiemar Germany, as the latter imploded spectacularly despite its many attractive second and third order qualities for its extreme failure of first-order qualities, while the former is quite possibly the single best state in first-order qualities standing today.
Though indications are that the balancing act involved is breaking down quickly, now that the problems of narrow-authority, low-freedom societies are rearing their head once more. A single mistake, from a very small organization, spells enormous damage to the vastly larger group. And that One Child Policy's demographic implosion is closing in, while the catchup benefits of the global economy run out and the debts are coming due, because that enormous power of central planning didn't look far enough ahead to predict its slowdown and be prepared for growth to need to give way to stability.
Keynesian Economics, lads, never forget to bank your gains today to cover for your losses tomorrow! The lack of this is the central flaw of modern economies, as a whole, since almost every particular thing bricks the moment things start going downhill for it.
