At some level there has to be an instinct for cooperation. It appears that intellectual greed overwhelms that in most cases.

Since all higher social animals show some forms of morality, which is to a degree certainly hard wired, the real question is. Why would humans be the exception ?

Everything has to be hard wired at the bottom of it. As Marionville below says. "Human survival from out earliest ancestor's time has relied on co-operation and reciprocity" But if we were not hard wired to value survival itself at the very least, then why bother with anything at all ? (Why bother eating and drinking, and why be afraid to end all our pains by jumping of a cliff ? )

And it would also I think be very hard for those (often theists) who like to argue there is no hard wired basis to morality, to argue well enough, to persuade most thinking people that we are not hard wired to value things like social status, which is often won through kindness and fair play, or that the emotional feelings like, pride, love, hate, pity and empathy for example are learned, because we all know that, for example, another's cry of pain triggers a purely emotional response in the first instant.

To that we then add the benefits of personal experience, we learn the things which earn us respect and social status, cultural experience, we learn culturally from the experiences of others solving the same problems, and reason, we can work out how much better it is to live in a society which knows and respects ideas like the golden rule.

Things like morality are complex and factors like culture, experience, reason, and hard wired emotions, weave together in complex knots , so that the whole structure is hard to see or untangle. This makes it easy for those who wish exploit others by using untruth, to use our natural laziness and fear of uncertainty, ( two other things that are hard wired), to persuade people to buy into their easy simple ready made answers, like, 'god gave us morality.'

Human survival from our earliest ancestors’ time has relied on co-operation and reciprocity. Societies developed from collective groups or tribes, and to be cast out from said groups meant certain death. That was a strong motivator to not transgress the rules and mores of the tribe. Stealing and killing within the group were therefore taboo, and from that understanding I believe morality sprung, It didn’t extend further than their own tribe of course, because they fought and killed other tribes and stole their property, enslaved them and generally behaved barbarically towards them in the pursuit of increasing their territory and preserving their own. . That has been the general template of human behaviour since earliest times and is still the case, even in the 21st century, only we have developed international laws and conventions to prevent this behaviour, not always adhered to by every country unfortunately.

In my opinion, the foundations of morality are evident in other species, particularly our closest relatives. And just what are the foundations of morality? I have come to believe that there are only two pillars (or core thinking processes) upon which ALL morality is based: 1) Empathy; and 2) Reciprocity. Humanity's excruciatingly detailed latticework of ethics and morays are founded on those two basic principles. Clearly, an evolutionary advantage was conferred to hominids who developed empathy and who adhered to reciprocity, or fairness.

There is evidence that some "morality" is hardwired. Instincts to protect young and instincts against killing are evident but can obviously be overridden. There is also likely to be an evolutionary aspect to ethics in society, in that societies which didn't have some particular rules don't survive. For instance the inhibition about incest in the general population is postulated to have come about through the destruction of societies that allowed it (if only because of the imbalance to family dynamics) although it is known to have been tolerated in royal families.

there's no "ethic gene"

I think Pinker tried to make something out of Peter Singer’s expanding circle in an arc spanning several books. Start with Hobbes’ war of all against all or Tennyson’s nature red in tooth and claw or Kant’s dictum: “Out of the crooked timber of humanity no straight thing was ever made.” Well people obviously care for themselves in an egoistic manner. And they care for family because implicitly shared heritage and recognize doing favors for less related others in immediate vicinity in smaller groups with reputation based on word of mouth may be reciprocated.

Slights work the same way which escalate into vendettas. Retributive justice is ugly.

Cooperation touted by scholars such as Kropotkin sounds better than pure selfishness but has dark implications considering through history such tendency remained parochial instead of being universalized beyond the ingroup. The question remains how does someone transcend interests of self and ingroup? And think of the dark political artwork Machiavelli related based on observations of behavior of popes and condottieri. That’s a kernel of human nature, but not justifiable as such. I go with Bulldog Huxley in that we must battle cosmic nature rather than emulate it.

I’m guessing both.

There is a genetic predisposition for social animals to behave more or less socially. That should be obvious. But there is also pressure from the group that discourages selfish, harmful behavior. Puppies can be observed to get in trouble when they get too rowdy. Their mamas nip their ears. Human toddlers have been seen getting swatted on the buttocks. Adult humans that cause harm can be observed to be locked up in cells.

There is a third possibility also: trans-personal awareness. In this mode individuality becomes softened. Individuals are guided by universal consciousness rather than by ego alone. An organism with trans-personal awareness acts in concert with others and is imbued with a sense of oneness.

Examples abound.

Well the precursors to morality are kin altruism and reciprocity and they seem largely inborn. And I wouldn’t equate them with morality proper as they each are insufficient, leading respectively to nepotism and cronyism without something breaking the mold of such ingroup biasing.

Morality in part (as explained below) is dependent upon rules/laws imposed upon someone and their ability or lack of ability to follow those rules/laws. The issue about this part of morality is that rules/laws can be arbitrary.
arbitrary
based on random choice or personal whim, rather than any reason or system.
"his mealtimes were entirely arbitrary"

(of power or a ruling body) unrestrained and autocratic in the use of authority.
"arbitrary rule by King and bishops has been made impossible"

With rules/laws being arbitrary this means that the fact of non-sense laws can affect the apparent morality level that a person has. I could argue that the traffic laws of a stop sign requiring a complete and total stop is at times not so hard set necessary for a driver given some situations. However, in view of such a law, the law itself does not care if the stop sign is out in the country 20 miles from any other stop sign and only 5 cars a day drive down the road. A local farmer living near the stop sign and not coming to a total complete stop ever time makes for the farmer to have technically a lower level of morality because the farmer "rolls" thru the stop sign because he can see for miles either way to know that no cars are crossing the intersection.

You say, "Richard Dawkins says that some people believe that they draw their morality from Science and some say you cannot draw morality form science."

Morality is defined as principles concerning the distinction between right and wrong or good and bad behavior.

Principle is defined as a fundamental truth or proposition that serves as the foundation for a system of belief or behavior or for a chain of reasoning. www.lexico.com

Moral comes from the Latin word mores, for habits. The moral of a story is supposed to teach you how to be a better person. If moral is used as an adjective, it means good, or ethical. If you have a strong moral character, you are a good member of society. If someone is a cheat and a liar, you might say, "She is not a moral person." [vocabulary.com]

"... draw their morality from Science" Science is the study of nature. People do learn things from observation of nature.

Moral/morality is viewing the actions, activity, habits, etc. of a person and then comparing those to 2 different things.

1. First, those things are compared to standards, rules, or laws that establishes 'RIGHTS AND WRONGS".

Example: The rules says: no chewing bubble gum while walking. Sue was seen chewing bubble gum while walking. Sue violated the rule. Sue would have low morals in view of the rule of "no chewing bubble gum while walking".

OR

2. Second, those things are compared to good and bad (evil).

Example: Sue walked while chewing bubble gum. While walking works out leg muscles it does not specifically work out facial muscles. Chewing gum gave the added benifit (good) for Sue to get her facial muscles worked out while walking. No one was harmed(suffered an evil) by the fact that Sue chewed bubble gum while walking. Sue would have high morals( good health habit) for chewing bubble gum while walking.

Sue was immoral for violation of the rule but was moral in view of a health benefit.

etymology moral(adj):
mid-14c., "associated with or characterized by right behavior," also "associated with or concerning conduct or moral principles" (good or bad), from Old French moral (14c.) and directly from Latin moralis "proper behavior of a person in society," literally "pertaining to manners," coined by Cicero ("De Fato," II.i) to translate Greek ethikos (see ethics) from Latin mos (genitive moris) "one's disposition," in plural, "mores, customs, manners, morals," a word of uncertain origin. Perhaps sharing a PIE root with English mood (n.1).

From late 14c. as "of or pertaining to rules of right conduct" (opposed to non-moral, amoral) and "morally good, in accordance with rules of right conduct" (opposed to immoral). Of persons, "habitually conforming to moral rules," 1630s. From 1680s with reference to rights, duties, etc., "founded on morality" (opposed to legal).

Applied to indirect effect in moral support (1823), moral victory (1888), where the notion is "pertaining to or affecting the character or conduct" (as distinguished from the intellectual or physical nature), a sense attested from 1590s; in this sense, compare morale. Related: Morally.

moral(noun):
"moral exposition of a story, the doctrine inculcated by a fable or fiction, the practical lesson which anything is designed to teach," c. 1500, from moral (adj.) and from French moral and Medieval Latin moralia. In this sense, morality was used from late 14c. The earlier noun use of moral was "a commandment pertaining to morals."

[etymonline.com]

I have often pondered the same questions. I don't have the answers, but the subjects that most fascinate me are the ones who appear to be nurtured one way and somehow (instinctively? intuitively? via external circumstances?) go the opposite direction.