I don't see how it's inconsistent logic to believe that some morals are inherent in most people, imo some morals could have been driven by natural selection, while many are just a product of culture, religion, etc. But I really can't say there's a scientific basis for that claim.

Overall, given what you've stated, I'd still lean towards morals being subjective. If you were to teach a human from birth to adulthood that having sex with a child is ok, and normal, they would probably not believe it to be morally wrong. On the flip side, things like sociopathy may also have been driven by selection, as it is a genetic condition. Most people wouldn't agree with a sociopath's morals, but they are objectively born with those tendencies. So....it's both, I don't get why that has to be inconsistent.