The question here isn’t whether secular humanists can be moral exemplars – on certain indices of morality, at least, of course they can be. The question here, rather, is whether secular humanism is true.
One approach with secular humanism is to say that, on an individual level, we want our society to become better, and so it makes sense to advocate for laws that will create this (and support punitive measures against those who transgress them). Among other things, this includes support for laws that might help to curb self-interest in favour of social interest. Hence, through self-interest we are able to generate a society that overcomes mere self-interest and instead creates social goods, and so tends to actually further self-interest.
This makes sense, and is shared by Christianity. The problem is that it doesn’t address the question of why on an individual level one should do what is right, in a specific instance, if it conflicts with one’s self-interest. For a common intuition is that there are some things we ought to do, even though they could conflict with our self-interest.
This problem leads to two further secular humanist approaches.
The first is to explain these sorts of moral intuitions by appeal to evolution. The basic idea is that we have intuitions which were selected for because they helped to get the individuals or societies who had those intuitions to survive and then reproduce, more than those who did not have those intuitions.
If our moral intuitions are just evolutionarily selected, though, for the purpose of increasing fitness, then why should we follow them in a particular, individual case if they conflict with our self-interest?
So, this second approach leads to a kind of nihilism – the view that there is no good reason to follow them – at the individual level – when they conflict with our self-interest. I.e., at best, the intuitions are rough guides to what might be good for us or our society (so see the first approach), but there is no good reason to follow them at the individual level in a specific instance if that means a conflict with our self-interest. (So, if we can get away with it, by breaking the law or what have you.)
The unsatisfactoriness of this kind of moralistic nihilism leads to a second approach, which is that there is an objective moral order reflected by these moral intuitions, and which applies individually. Here, our evolutionary intuitions for some reason do get close to this objective order in at least some cases, and we ought to follow them at the individual level. The reason to follow them is not because of self-interest but simply because it is right – and so posits that what is right is intrinsically worthwhile from an individual perspective.
The basic problem, though, from a secular perspective is that it posits an objective moral order that is relevant in a direct, personal way even when it conflicts with self-interest. I.e., where did this moral order come from? What is its nature?
Typically, these questions are left unanswered. However, one of the chief supposed benefits of the secular humanist approach (not positing something like God) now comes partially undone, because this objective moral order is getting pretty close to something like aspects of the Christian God – yet without the contextual conceptual framework that situates and says why this exists, as Christianity attempts to do.
So, it doesn’t seem like there’s a satisfactory picture of morality in secular humanist terms.