There is a strong tradition among evolutionary biologists and psychologists that Homo sapiens is a species that is not only capable of acting on hidden selfish motives, they claim that we’re designed to do it. According to this theory, our brains are built to act in our self-interest while at the same time trying hard not to appear selfish in front of other people.

While there can be no doubt that humans are acting selfishly often enough, I do not think that selfishness is the human default. There is some evidence that our brains are 'social brains' but not primarily Machiavellian brains.
Here is my evidence against the theory that H. sapiens is basically another selfish ape:

Exhibit A: Moral emotions.
In a society of Machiavellian apes like chimps the last thing that would evolve are emotions like shame, guilt, remorse. What would be expected to evolve is a range of submissive gestures (as we can observe with chimps). If I have angered alpha and alpha is mad at me, it is useful to signal „it wasn't meant to offend you! I'm your humble servant! Please don't bite!“ But it would be useless or even harmful to actually FEEL shame or remorse, because such feelings are an obstacle to scheming and cunning, even when nobody is watching. But natural selection favored the evolution of these moral emotions.

Exhibit B: The white sclera of the human eye.
All other apes have a dark sclera with makes it diffcult for others to guess what an individual is looking at, which is quite useful when a subordinate male is leering at the juicy posterior of a female that 'belongs' to alpha..... If humans were just smart and scheming, cunning.. in a word: Machiavellian apes, the last thing natural selection would have favored is an eye that gives away the intention of an individual. The white sclera is a trait that o n l y makes sense in a society where people should be able to read each others mind, in order to join attention and to facilitate cooperation (see Tomasellos seminal book „A Natutal History of Human Morality“ for details. It is a trait that is group-beneficial. Hiding intentions? Deceiving? Difficult with a white sclera.

Exhibit C: Norm psychology and 'third-party punishment':
My neighbor Susan beats her child, but this should be no skin off my nose. But I care. Real people do care. Self-interest alone doesn’t explain how people operate in real human societies—we judge norm violators and often punish them even it's costly for us, because we identify with those norms.

Exhibit D: Trust bias.
The default option when we deal with strangers is to trust them. If we were just selfish Machiavellian apes, the default option would be "I am suspicious until you have proven that you are trustworthy“. But that is not how it works. It is the other way round: "I trust you until there is evidence that you are not trustworthy." - This basic trust is the 'sine qua none' of any society beyond the size of a band, it is what allows trade , or what allows us to take our money to banks. Without a 'trust bias' there simply would be no trade or banks, because my first thought would be "This guy may smile at me, but he just wants my money to walk off with it." The reason why fraudulent 'sellers' on ebay and other con men like Bernie Madoff are so successful is our hard-wired propensity to trust strangers. In a Machiavellian world where everybody is constantly trying to deceive, such a trust bias could never have evolved.

Exhibit E: Human ultrasociality:
Humans are the only vertebrate with large-scale cooperation among non-related individuals. We are able of feats of cooperation only rivaled by eusocial insects like bees or ants. But in their case, it is a family business. We have built cities and states cooperating with total strangers. The operating system behind this feat is provided by norms. If Machiavellism were the core of the 'conditio humana', no city would ever have been built because people would have quarreled all the time, would have tried to free-ride and to outwit each other, and would finally have dispersed like people in the Bible did when they tried to build the Tower of Babylon, but God confused their language.

Just imagine a truly Machiavellian world in nuce, like (say) the White House under Trump, where everybody is constantly trying to deceive, to scheme, to gain some personal advantage, to hide his or her intentions, to form only tactical alliances, only to be abandoned after the 'partner' is no longer useful... Does anybody think that such a bunch of selfish and cunning individuals could ever achieve anything together, like, say, building an irrigation system, or a fish weir, or something really big like Göbekli Tepe? Of course not. They wouldn't even survive a month out in the wilderness, where everyone depends on each other. Where only groups can prosper that manage to function together.

But that's exactly what Mother Nature aka evolution did with our ancestors (Homo habilis, erectus, and so on...): Natural selection formed a naked ape, where the individual itself was helpless, frail, an easy prey for predators, but all those shortcomings and frailties were more than compensated by basically two specifically human traits:
(-a-) ultrasociality, a unique form of groupishness, with the ability to form functional groups consisting of very good team-players, and
(-b-) cumulative culture, the ability not to hide useful knowledge as a means to outwit and out-compete my competitors, but to pool and transmit useful knowledge for the benefit of all.