Humans and melanin.
One of the claims in the OP was that humans are not primates as our skin is "not adapted to sunlight exposure like all other primates".
There is twofold wrongness about this particular claim. In fact, its back to front in its wongness.
Firstly, human skin is more adapted to dealing with sunlight that that of other primates, not less. Non-human primates generally have heavy fur which protects their skin from direct UV exposure. However, all primates have greater or lesser amounts of melanocytes, which produce melanin which help to absorb UV radiation and reduce the damage done to the dermis.
Humans adapted to living in equatorial areas, exposed to the greatest amounts of UV radiation, generally have more melanocytes in their skin than humans adapted to living in much higher/much lower latitudes. Up to 40 times more, in fact. For, example, according the the US Cancer Council, African America skin has a sun protection factor of 13.4, compared with 3.4 in white skin.
In the early 1980s Jablonski and Chaplin used NASA UV models, along with information on dietary intake of Vitamin D, to successfully predicted the skin colors of indigenous people across the globe based on where they lived. Heavy pigmentation developed in human skin as an adaptation during transition from an aboreal habitat to a plains/savannah habitat. Light skin only evolved as humans moved out of Africa into higher latitudes.
Secondly, other primates also produce melanin. In fact, the range of melanin production other primates is incredibly diverse.
Some primates have high melanin levels at birth - when they have little or no fur - but lose most of their melanin production facilities during development, as their fur grows heavier. In contrast, chimpanzees and certain macaque and other monkey species have almost no melanin at birth, but are primarily black-skinned in adulthood.
There are also occurrences of white-skinned chimpanzees. This suggests that melanin levels are mostly irrelevant to most of the rest of the primate order due to their thick fur, apart from a few key areas such as the face, hands and tops of feet.
Thus, humans, without heavy fur and with high levels of melanin production depending on where they live, are better adapted to sunlight exposure than other primates.