Whereas I see you as having 'faulty thinking'. I don't see how you conclude that a person who does not openly state theological reasons for their ethical position had no religious (as well as non-religious) reasons for reaching that position. I know when I talk to people I adapt my conversation according to my audience, so I won't bring Islam into the conversation if I am talking to non-Muslims even if my interpretation of Islam formed part of my reasoning in arriving at an ethical position.
I'm not talking about people who 'adapt' their conversation according to their audience, but people who adapt their conversation according to the topic, in this case the ethical issue. In other words people who are happy to be very clear that they are christian and their views on some topics are linked to christian theology, yet on other ethical topics the notion of christian theology is totally absent from their justification with reliance on other ethical approaches - so extolling the virtues of the humanist golden rule, or a nod to good old consequentialism, perhaps alluding to the kantian categorical imperative.
Sure I understand that some christians don't like to talk about their religion with others, particularly non christians, but plenty do. So why would they make it clear that some of their ethical reasoning is linked to christianity, but others not, unless that actually represented their reasoning.
Sure I understand there are some people who are completely 'pure' in their ethical positions, using theology or an individual secular ethical theories to govern their views. But there are also plenty of others for whom their ethical thinking is driven by a kind of pick and mix of ethical approaches that are prevalent in our society, including religious and non religious approaches.
From my own experience (and that includes over a decade of teaching ethics) that many (if not most) people use a hybrid of approaches to inform their ethical positions. And that mixed approach is ultimately driven by the complex range of ethical positioning that is deeply embedded in our societies and families with our own upbringing, familial/societal influence plus experiences in life being instrumental in the ethical positions people adopt and the underpinning ethical theories that drive those positions. Now, of course, most people aren't overtly aware that their thinking is kantian, or consequentialist, or humanist, or even overtly religious - but those elements are just under the surface of ethical reasoning - why because they are so deeply embedded in the fabric or our society.