You're generalizing and obfuscating too much, as only some religious people even claim to know there is an objective morality, let alone that everyone should follow the same one, and nonreligious people have argued multiple sides in the past.

Some nonreligious say that there is no objective morality, but that we should try to formulate a generally agreed upon one to make life better and happier, whereas others argue that there is a form of objective morality built into humans through evolution but can be ignored, arguing that religion and others have distorted it over time to achieve their own ends. Then there are some who think morality is completely subjective and doesn't really matter as a whole, or that we can never perfectly communicate that morality to others so it's pointless to try, along with many other ideas.

Many organized religions do say they have the only true morality given to them by God, but other religions don't, along with individuals in each religion having their own opinion.

My point being that your argument is too simplistic to be accurate.