The comparison to health is a red herring as there are not extra ‘competing axioms’ of health and any 'undecidability' of the healthiness of an action can be, in principal, solved by having better knowledge of the physical world. In contrast, there are many competing moral axioms that cannot be decided by reference to evidence. Indeed there is great doubt as to whether moral axioms are meaningful or whether moral statements can be truth bearing. Certainly the opposite cases cannot be proven. Also, there are plenty of 'hard cases' which can be easily quoted which demonstrate the difficulty of deciding between moral axioms. As far as I am aware, there are no 'hard cases' of health that cannot in principle be decided by science alone.
That you no longer are capable of even recognizing that 'health' is analogous is perhaps the best argument in favor of Harris' proposal. Once 'health' is wrested from the grip of teleology, it doesn't even occur to us to question the idea that one can make reference to the physical world in order to answer health questions.
Also, there are particular single actions that are unhealthy for everyone, e.g. being vaporised in a nuclear blast, and therefore there is a clear objective basis for health.
Yet we are unable to tell whether being vaporized in a nuclear blast would be perceived as good for everyone? I find that a bit hard to believe.
There is also another massive problem for Harris which magnifies all his other problems. Harris says that the well-being of conscious creatures must be the basis for deciding values. Yet he does not seem to provide a working definition of what consciousness entails or a justification of his definition as a dividing line in terms of well being. If we assume that Harris has a broad definition in mind, simply ‘the capacity to feel well-being or otherwise’, we must include the well-being of all conscious creatures in the entire universe into our ‘worst possible misery for everyone’ formula. If it wasn’t bad enough already, perhaps if we start removing Harris' anthropocentric arguments and replace the words ‘human’ and everyone’ with ‘all conscious creatures in the universe’, we can see how distorted, truly subjective and meaningless his supposed objective basis for morality becomes. We have to start wondering what the worst possible misery for individual tadpoles looks like, if they have the capacity to feel pain and how much tadpole worst possible misery equals one human worst possible misery, (if we presume that all human worst possible misery is an equal amount of misery, which is almost certainly either meaningless, undecidable or wrong). As anyone should see, this is only going to lead to truths of the most subjective kind.
I think this is your strongest argument against the idea that Harris is proposing utilitarianism - i.e. you have shown that his proposals fail to be meaningful or provide some of the necessary information if you try to force them into that particular slot.
Linda