I am pretty sure that ethics is a part of everybody's lives on a day to day basis e.g. making decisions about whether to cheat at cards or taxes, or contribute money to cause x, and that in a predominantly non-religious society like ours people make these decisions without consulting God, clergy, ethicists, neurobiologists or psychologists...
I'm not totally sure how they make these decisions except that is clearly driven by that innate moral sense which is also steered and possibly co-opted by early learning, exposure to religion, peers, mood, what they saw on TV last night... a long and arbitrary list.
The newspapers are filled with ethical dilemmas and debates – and in addition, many policy questions involve an efficiency vs. equity trade-off that has to be viewed in both economic and ethical terms. These type of issues are generally resolved without involvement of professional ethicists of any stripe.
There are some issues where clergy clearly seem to get the floor – maybe euthanasia for example, but I don’t know if they have the floor exclusively, with op-eds and 60 Minutes type shows usually putting both sides (if only to make things more juicy!)
It seems that the point is at least in part around who has "voice" in our culture and our media around tricky issues such as genetic testing of embryos, abortion, gay marriage etc.
My hope is to find Richard Dawkins or Sam Harris or many of the other thinkers that I admire in many ways taking on more controversial topics: current wars the West is involved in, peak oil, excessive consumption, and third world debt.
These are all more complex issues than whether or not God says you can marry a 12 year old against her will, and the underlying ideologies, while not religious (except possibly the Iraq war) deserve to be unpacked and attacked in the same way these religious questions are attacked. That is, if the real concern is a more moral society and not just a knee jerk response to a bearded dude who claims to know God's will.
As you say, ours is a secular society, but non-religious people are still a minority. The "religious issues" such as genetic testing of embryos, abortion, and gay marriage actually affect all of the population- and I believe that the imposition of majority views upon minority rights make these civil rights issues. I think that the complex issues that you raise are important, but we should begin with these simple "low-hanging fruit" issues first.
As far as the other issues are concerned, I believe that those are religious/atheist issues too. I don’t know if you have met many religious environmentalists, but in Genesis it says that man was given dominion over the earth- and it has been found that biblical literalism has a negative correlation with willingness for environmental spending (Greeley 1993).
This indicates that faith in God’s provenance is often used as a solution to problems of this kind. Utah’s house of representatives has passed a bill saying that AGW is a hoax, and many religious groups claim that environmentalism is a destructive new religion.
It is important to realise that the New Atheists are rational and mainly liberal, and are likely to hold progressive views, but are focusing their efforts on what they see to be the root of the problem. I’m inclined to follow their lead, having had many unproductive debates on the complex issues- usually with people who have a limited willingness to listen to evidence.
Your model of the innate moral sense seems to illustrate the importance of the moral influences that we are exposed to. If this is true, then the matter of religion having ‘the voice’ becomes an important one, especially if we can argue that they unfairly claim authority for their stances.