I agree, and I think this is one of the benefits of Christianity to other forms of theism. Not only did God tell us how we “should” behave, but he gave us an example in the form of a man who (if you are a Christian) exemplifies what it means to live this out, in a way we as humans can relate to.
On one hand, I agree that saying “because God says so” is not a satisfying answer. On the other hand, if God created all this stuff and also provided a set of guidelines of how he would like us to treat other people (with tangible benefits to ourselves and others when we do so), then it makes sense why we should care about what God declares is moral. We are free to choose not to follow these directives of course. If the commands of God follows our moral intuition and gives us a basis for justifying it, then I think we have good reason to invoke God in morality.
Ayer and the other positivists claim that non-falsifiable propositions are meaningless/not grounded in reality. Whether moral claims are factual (I.e. there really is an obligation or teleology for humans) is very important. I am not denying that harm, fairness, human wants and needs are grounded in reality. What is difficult to justify is the content of a claim that we have an “obligation” to change our own behavior/wants/needs.
An obligation or duty is a different kind of thing. It involves something intended to change our behavior from how it “is” to how it “should” be. For instance, when I see an animal eating another animal, I understand that there is animal suffering going on. At the same time, I may not feel an obligation to step in and “save” the animal suffering. If I see a human hurting another human, I may feel an obligation to step in and help. Is there a “right” response, in that I should/should not feel such an obligation, or actually intervene?
When someone says “people should step in and help others,” they often want this to be taken as a proposition or a statement that expresses truth. Now this “should” is ether relative or objective. It is relative, the content of someone’s statement is really “I would prefer it if people step in and help others.” After all, the other person is free to have their own relative “should.” Under relativity, these both seem (to me) to be equally epistemically valuable, despite me “liking” one more than the other.
Alternatively, if such a “should” is not relative and is instead ontologically grounded, then such a statement can be objective and even a true proposition. A universal obligation makes a lot of sense with the existence of God. God’s character forms the ontological status for commands which are an expression of divine will. Since God created everything, he has a “right” to say how we “ought” to behave or what good functioning people behave like. When we make moral claims about the “rightness” or “wrongness” of things, we can make these claims with reference to principles that exist outside of human thought. If God exists and cares about how we treat each other, then the idea of a duty or obligation makes a lot of sense: it comes from an agent with a will and desires. This avoids the issue of obligations being relative to each culture and thus “cancelling out.”