This is where his objection to hell becomes a bit of a contradiction. Christian objective morality is objective precisely because simply ignoring it has consequences. It has supernatural consequences. If it had only natural consequences you might say it is not a moral principle at all but rather a principle of cause and effect. So a principle like:
People should not lie or else others will not trust themThat is not really moral. That is just a principle of logic. If you do A then B will result. It does not actually assert that B is bad. It just says A causes B so if B is bad then A should be avoided. So this is more of a statement of social science than morality.
What you need is more than statements about how to get certain results in human relations. Moral principles need to go beyond natural, measurable results. It needs to make precise what man desires when he wants to be good. Suppose you did that. You spelled out precisely what was good and what was evil. Why should we care? Does there not need to be some reason to pursue good and avoid evil?
That makes me wonder about his scoffing at the idea of heaven and hell. Has he not made his whole objective morality irrelevant? If there is no equivalent of heaven and hell then all you can get for being good are the things that naturally flow from being good. In other words you are back to the principles of cause and effect. You might also get a gold star from Prof. Fincke as well but mostly you are hoping to get some pain and pleasure benefit in this life.
You might argue that a particular moral system will give you better results long term than if you used other non-moral decision-making methods. But that would just be true until a better method came along. If we could get a more accurate cause and effect model then we could ditch morality. Again that does not seem like a moral system but just an indirect way of getting the best result for me.
Only a supernatural reality can explain why we should care about good and evil regardless of their natural consequences. Once you dismiss that possibility then you are down to what people subjectively see as good and evil. A goodness that people are willing to give up something material get. As soon as someone rejects the moral principle then it no longer applies to them. It might be stupid for them to ignore it but it would not be immoral.
Lastly, you wonder how he can every argue for a morality in a way that is significantly more convincing that people's arguments for God. I can see people dismissing arguments for God. It is not hard to be hyper-skeptical and demand absolute proof for every premise. But if you want to turn around and argue that some objective morality does exists then you are in trouble. This same skepticism will shoot down any argument you try and make.
If you do get past that and accept that virtues exist, or some such moral entities, then you need to explain how they came to be, why we should order our lives towards them, and how we can know what they are. This was the path that led Leah Libresco to the Catholic faith.